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## Unit: Matrices

- Intro to matrices (Opens a modal)
- Matrix dimensions 4 questions Practice
- Matrix elements 4 questions Practice

## Representing linear systems of equations with augmented matrices

- Representing linear systems with matrices (Opens a modal)
- Represent linear systems with matrices 4 questions Practice

## Elementary matrix row operations

## Row-echelon form & Gaussian elimination

## Adding & subtracting matrices

- Adding & subtracting matrices (Opens a modal)
- Add & subtract matrices 4 questions Practice
- Matrix equations: addition & subtraction 4 questions Practice

## Multiplying matrices by scalars

- Multiplying matrices by scalars (Opens a modal)
- Multiply matrices by scalars 4 questions Practice
- Matrix equations: scalar multiplication 4 questions Practice

## Properties of matrix addition & scalar multiplication

- Intro to zero matrices (Opens a modal)
- Properties of matrix addition (Opens a modal)
- Properties of matrix scalar multiplication (Opens a modal)

## Multiplying matrices by matrices

- Intro to matrix multiplication (Opens a modal)
- Multiplying matrices (Opens a modal)
- Multiply matrices 4 questions Practice

## Properties of matrix multiplication

- Defined matrix operations (Opens a modal)
- Matrix multiplication dimensions (Opens a modal)
- Intro to identity matrix (Opens a modal)
- Intro to identity matrices (Opens a modal)
- Dimensions of identity matrix (Opens a modal)
- Is matrix multiplication commutative? (Opens a modal)
- Associative property of matrix multiplication (Opens a modal)
- Zero matrix & matrix multiplication (Opens a modal)
- Properties of matrix multiplication (Opens a modal)
- Using properties of matrix operations (Opens a modal)
- Using identity & zero matrices (Opens a modal)

## Matrices as transformations

- Transforming vectors using matrices (Opens a modal)
- Transforming polygons using matrices (Opens a modal)
- Matrices as transformations (Opens a modal)
- Matrix from visual representation of transformation (Opens a modal)
- Visual representation of transformation from matrix (Opens a modal)
- Use matrices to transform 3D and 4D vectors 4 questions Practice
- Transform polygons using matrices 4 questions Practice
- Understand matrices as transformations of the plane 4 questions Practice

## Determinant of a 2x2 matrix

## Introduction to matrix inverses

- Intro to matrix inverses (Opens a modal)
- Determining invertible matrices (Opens a modal)
- Determine inverse matrices 4 questions Practice
- Determine invertible matrices 4 questions Practice

## Finding the inverse of a matrix using its determinant

## Practice finding the inverses of 2x2 matrices

## Determinants & inverses of large matrices

- Determinant of a 3x3 matrix: standard method (1 of 2) (Opens a modal)
- Determinant of a 3x3 matrix: shortcut method (2 of 2) (Opens a modal)
- Inverting a 3x3 matrix using Gaussian elimination (Opens a modal)
- Inverting a 3x3 matrix using determinants Part 1: Matrix of minors and cofactor matrix (Opens a modal)
- Inverting a 3x3 matrix using determinants Part 2: Adjugate matrix (Opens a modal)
- Determinant of a 3x3 matrix 4 questions Practice
- Inverse of a 3x3 matrix 4 questions Practice

## Solving equations with inverse matrices

- Representing linear systems with matrix equations (Opens a modal)
- Matrix word problem: vector combination (Opens a modal)
- Use matrices to represent systems of equations 4 questions Practice

## Model real-world situations with matrices

## About this unit

## Free Mathematics Tutorials

Matrices with examples and questions with solutions.

Examples and questions on matrices along with their solutions are presented .

## Definition of a Matrix

Example 1 The following matrix has 3 rows and 6 columns.

## Matrix entry (or element)

- Add, Subtract and Scalar Multiply Matrices
- Multiplication and Power of Matrices
- Linear Algebra
- Row Operations and Elementary Matrices
- Matrix (mathematics)
- Matrices Applied to Electric Circuits
- The Inverse of a Square Matrix

## Popular Pages

Resource links, share with us.

## Matrices and Determinants: Problems with Solutions

## Solving Systems of Linear Equations Using Matrices

## The Example

One of the last examples on Systems of Linear Equations was this one:

## Example: Solve

We went on to solve it using "elimination", but we can also solve it using Matrices!

But first we need to write the question in Matrix form.

## In Matrix Form?

OK. A Matrix is an array of numbers:

Well, think about the equations:

They could be turned into a table of numbers like this:

We could even separate the numbers before and after the "=" into:

Now it looks like we have 2 Matrices.

In fact we have a third one, which is [x y z] :

Why does [x y z] go there? Because when we Multiply Matrices we use the "Dot Product" like this:

Try the third line for yourself.

## The Matrix Solution

Then (as shown on the Inverse of a Matrix page) the solution is this:

So let's go ahead and do that.

First, we need to find the inverse of the A matrix (assuming it exists!)

Using the Matrix Calculator we get this:

(I left the 1/determinant outside the matrix to make the numbers simpler)

Then multiply A -1 by B (we can use the Matrix Calculator again):

And we are done! The solution is:

Just like on the Systems of Linear Equations page.

Quite neat and elegant, and the human does the thinking while the computer does the calculating.

## Just For Fun ... Do It Again!

For fun (and to help you learn), let us do this all again, but put matrix "X" first.

So we will solve it like this:

Then (also shown on the Inverse of a Matrix page) the solution is this:

This is what we get for A -1 :

In fact it is just like the Inverse we got before, but Transposed (rows and columns swapped over).

## Definition, Formulas, Solved Example Problems - Solved Example Problems on Applications of Matrices: Solving System of Linear Equations | 12th Mathematics : UNIT 1 : Applications of Matrices and Determinants

Solution to a System of Linear equations

## (i) Matrix Inversion Method

Solve the following system of linear equations, using matrix inversion method:

5 x + 2 y = 3, 3 x + 2 y = 5 .

Then, applying the formula X = A −1 B , we get

So the solution is ( x = −1, y = 4).

## Example 1.23

Solve the following system of equations, using matrix inversion method:

The matrix form of the system is AX = B,where

So, the solution is ( x 1 = 1, x 2 = 2, x 3 = −1) .

Writing the given system of equations in matrix form, we get

Hence, the solution is ( x = 3, y = - 2, z = −1).

## (ii) Cramer’s Rule

Solve, by Cramer’s rule, the system of equations

x 1 − x 2 = 3, 2 x 1 + 3 x 2 + 4 x 3 = 17, x 2 + 2 x 3 = 7.

First we evaluate the determinants

So, the solution is ( x 1 = 2, x 2 = - 1, x 3 = 4).

## Example 1.26

Hence the ball went for a super six and the Chennai Super Kings won the match.

## (iii) Gaussian Elimination Method

Solve the following system of linear equations, by Gaussian elimination method :

4 x + 3 y + 6 z = 25, x + 5 y + 7 z = 13, 2 x + 9 y + z = 1.

Transforming the augmented matrix to echelon form, we get

The equivalent system is written by using the echelon form:

Substituting z = 2, y = -1 in (1), we get x = 13 - 5 × (−1 ) − 7 × 2 = 4 .

So, the solution is ( x =4, y = - 1, z = 2 ).

## Example 1.28

Since v (3) =64, v (6) = 133 and v (9) = 208 , we get the following system of linear equations

We solve the above system of linear equations by Gaussian elimination method.

Writing the equivalent equations from the row-echelon matrix, we get

9a + 3b + c = 64, 2b + c = 41, c= 1.

So, we get v (t) = 1/3 t 2 + 20t + 1.

Hence, v (15) = 1/3 (225) + 20(15) + 1 = 75 + 300 + 1 = 376

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## Matrix Multiplication Questions

Also check: Important questions for class 12 Matrices .

Know more about the properties of matrix multiplication .

## Matrix Multiplication Questions with Solutions

Let us solve some questions to practise matrix multiplications.

Find the product of the following matrices:

Prove that for the matrices A and B, (A + B) 2 ≠ A 2 + 2AB + B 2 where

From (i) and (ii), it is proved that (A + B) 2 ≠ A 2 + 2AB + B 2 .

For given matrices A and B, find AB and BA, also prove that AB ≠ BA.

If the matrices S and Y are multiplication conformable, then k = 3.

If the matrices SY and WY are addition conformable, then p = n.

Verify the associative property for multiplication of matrices:

We need to prove that A(BC) = (AB)C.

From (i) and (ii) we get that A(BC) = (AB)C.

Show that the product of the two matrices

It is a null matrix when the difference between both angles is an odd multiple of 𝜋/2.

The product of both the matrix is given by

Let the given matrices A, B and C be

Find a matrix D such that CD – AB = 0

\(\begin{array}{l}Let,\:\:D=\begin{bmatrix}a & b \\c & d \\\end{bmatrix}\end{array} \)

We get the following equations,

Solving (i) and (ii) we get, a = – 19/2 and c = –3/2

Solving (iii) and (iv) we get, b = –9 and d = –1.

\(\begin{array}{l}A=\begin{bmatrix}1 & 2 \\2 & 1 \\\end{bmatrix}\end{array} \)

Let f(x) = 2x 2 – 3x, find det [f(A)] if

\(\begin{array}{l}A=\begin{bmatrix}-2 & 1 \\0 & 3 \\\end{bmatrix}\end{array} \)

Given, f(x) = 2x 2 – 3x, then f(A) = 2A 2 – 3A

\(\begin{array}{l}\Rightarrow f(A)=\begin{bmatrix}14 & -1 \\0 & 9 \\\end{bmatrix}\end{array} \)

Show that f(x).f(y) = f(x + y) where

## Recommended Videos

## Practice Questions on Matrix Multiplication

1. Find the product of the following matrices:

3. Verify the distributive property for multiplication over addition for the matrices:

4. For the given matrices A and B, verify (A + B) 2 = A 2 + B 2 , where

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## Matrix Calculator

Solve matrix operations and functions step-by-step.

- Add, Subtract
- Multiply, Power
- Determinant
- Minors & Cofactors
- Characteristic Polynomial
- Gauss Jordan (RREF)
- Row Echelon
- LU Decomposition New
- Eigenvalues
- Eigenvectors
- Diagonalization
- Exponential
- Scalar Multiplication
- Dot Product
- Cross Product
- Scalar Projection
- Orthogonal Projection
- Gram-Schmidt

## Most Used Actions

- \begin{pmatrix}3 & 5 & 7 \\2 & 4 & 6\end{pmatrix}-\begin{pmatrix}1 & 1 & 1 \\1 & 1 & 1\end{pmatrix}
- \begin{pmatrix}11 & 3 \\7 & 11\end{pmatrix}\begin{pmatrix}8 & 0 & 1 \\0 & 3 & 5\end{pmatrix}
- \tr \begin{pmatrix}a & 1 \\0 & 2a\end{pmatrix}
- \det \begin{pmatrix}1 & 2 & 3 \\4 & 5 & 6 \\7 & 8 & 9\end{pmatrix}
- \begin{pmatrix}1 & 2 \\3 & 4\end{pmatrix}^T
- \begin{pmatrix}1 & 2 & 3 \\4 & 5 & 6 \\7 & 2 & 9\end{pmatrix}^{-1}
- rank\:\begin{pmatrix}1 & 2 \\3 & 4\end{pmatrix}
- gauss\:jordan\:\begin{pmatrix}1 & 2 \\3 & 4\end{pmatrix}
- eigenvalues\:\begin{pmatrix}6&-1\\2&3\end{pmatrix}
- eigenvectors\:\begin{pmatrix}6&-1\\2&3\end{pmatrix}
- diagonalize\:\begin{pmatrix}6&-1\\2&3\end{pmatrix}

## Frequently Asked Questions (FAQ)

How do you multiply two matrices together.

## What is matrix used for?

## What is a matrix?

## How do you add or subtract a matrix?

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## How to Solve Matrices

Last Updated: October 7, 2022 References

## Setting up the Matrix for Solving

- If you have more variables, you will just continue the line as long as necessary. For example, if you are trying to solve a system with six variables, your standard form would look like Au+Bv+Cw+Dx+Ey+Fz =G. For this article, we will focus on systems with only three variables. Solving a larger system is exactly the same, but just takes more time and more steps.
- Note that in standard form, the operations between the terms is always addition. If your equation has subtraction instead of addition, you will need to work with this later my making your coefficient negative. If it helps you remember, you can rewrite the equation and make the operation addition and the coefficient negative. For example, you can rewrite the equation 3x-2y+4z=1 as 3x+(-2y)+4z=1.

- For example, suppose you have a system that consists of the three equations 3x+y-z=9, 2x-2y+z=-3, and x+y+z=7. The top row of your matrix will contain the numbers 3,1,-1,9, since these are the coefficients and solution of the first equation. Note that any variable that has no coefficient showing is assumed to have a coefficient of 1. The second row of the matrix will be 2,-2,1,-3, and the third row will be 1,1,1,7.
- Be sure to align the x-coefficients in the first column, the y-coefficients in the second, the z-coefficients in the third, and the solution terms in the fourth. When you finish working with the matrix, these columns will be important in writing your solution.

## Learning the Operations for Solving a System with a Matrix

- You will be working with some basic operations to create the “solution matrix.” The solution matrix will look like this: [6] X Research source
- Notice that the matrix consists of 1’s in a diagonal line with 0’s in all other spaces, except the fourth column. The numbers in the fourth column will be your solution for the variables x, y and z.

- It is common to use fractions in scalar multiplication, because you often want to create that diagonal row of 1s. Get used to working with fractions. It will also be easier, for most steps in solving the matrix, to be able to write your fractions in improper form, and then convert them back to mixed numbers for the final solution. Therefore, the number 1 2/3 is easier to work with if you write it as 5/3.
- For example, the first row (R1) of our sample problem begins with the terms [3,1,-1,9]. The solution matrix should contain a 1 in the first position of the first row. In order to “change” our 3 into a 1, we can multiply the entire row by 1/3. Doing this will create the new R1 of [1,1/3,-1/3,3].
- Be careful to keep any negative signs where they belong.

- You can use some shorthand and indicate this operation as R2-R1=[0,-1,2,6].
- Recognize that adding and subtracting are merely opposite forms of the same operation. You can either think of adding two numbers or subtracting the opposite. For example, if you begin with the simple equation 3-3=0, you could consider this instead as an addition problem of 3+(-3)=0. The result is the same. This seems basic, but it is sometimes easier to think of a problem in one form or the other. Just keep track of your negative signs.

- 1. Create a 1 in the first row, first column (R1C1).
- 2. Create a 0 in the second row, first column (R2C1).
- 3. Create a 1 in the second row, second column (R2C2).
- 4. Create a 0 in the third row, first column (R3C1).
- 5. Create a 0 in the third row, second column (R3C2).
- 6. Create 1 in the third row, third column (R3C3).

- Create a 0 in the second row, third column (R2C3).
- Create a 0 in first row, third column (R1C3).
- Create a 0 in the first row, second column (R1C2).

## Putting the Steps Together to Solve the System

- Copy down the unaffected row 3 as R3=[1,1,1,7].
- Be very careful with subtracting negative numbers, to make sure you keep the signs correct.
- For now, leave the fractions in their improper forms. This will make later steps of the solution easier. You can simplify fractions in the final step of the problem.

- Notice that as the left half of the row starts looking like the solution with the 0 and 1, the right half may start looking ugly, with improper fractions. Just carry them along for now.
- Remember to continue copying the unaffected rows, so R1=[1,1/3,-1/3,3] and R3=[1,1,1,7].

- Notice that the fractions, which appeared quite complicated in the previous step, have already begun to resolve themselves.
- Continue to carry along R1=[1,1/3,-1/3,3] and R2=[0,1,-5/8,27/8].
- Notice that at this point, you have the diagonal of 1’s for your solution matrix. You just need to transform three more items of the matrix into 0’s to find your solution.

## Verifying Your Solution

## Expert Q&A

## Video . By using this service, some information may be shared with YouTube.

## You Might Also Like

- ↑ https://www.mathsisfun.com/algebra/matrix-introduction.html
- ↑ https://www.mathsisfun.com/algebra/systems-linear-equations-matrices.html
- ↑ https://www.khanacademy.org/math/precalculus/precalc-matrices/representing-systems-with-matrices/a/representing-systems-with-matrices
- ↑ https://www.cuemath.com/algebra/solve-matrices/
- ↑ https://www.khanacademy.org/math/precalculus/precalc-matrices/multiplying-matrices-by-scalars/a/multiplying-matrices-by-scalars
- ↑ https://people.richland.edu/james/lecture/m116/matrices/operations.html
- ↑ https://www.math.tamu.edu/~dallen/m640_03c/lectures/chapter2.pdf
- ↑ https://www.varsitytutors.com/hotmath/hotmath_help/topics/solving-matrix-equations
- ↑ https://math.libretexts.org/Bookshelves/Algebra/Intermediate_Algebra_(OpenStax)/04%3A_Systems_of_Linear_Equations/4.06%3A_Solve_Systems_of_Equations_Using_Matrices
- ↑ https://www.cliffsnotes.com/study-guides/algebra/algebra-ii/linear-equations-in-three-variables/linear-equations-solutions-using-matrices-with-three-variables

## About This Article

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## What are Matrices?

## Matrix Definition

## Notation of Matrices

## Calculate Matrices

## Addition of Matrices

Subtraction of matrices, scalar multiplication, multiplication of matrices.

Properties of scalar multiplication in matrices

To obtain the element \(a_{11}\) of AB, we multiply \(R_1\) of A with \(C_1\) of B :

To obtain the element \(a_{12}\) of AB, we multiply \(R_1\) of A with \(C_2\) of B:

To obtain the element \({{a}_{21}}\) of AB, we multiply \(R_2\) of A with \(C_1\) of B:

Proceeding this way, we obtain all the elements of AB.

Properties of Matrix Multiplication

- A(BC) = (AB)C
- A(B + C) = AB + AC
- (A + B)C = AC + BC
- A\(I_m\) = A = AI n , for identity matrices I\(_m\) and I n .
- A\(_{m\times n}\)O\(_{n\times p}\) = O\(_{m\times p}\), where O is a null matrix.

## Transpose of Matrix

Properties of transposition in matrices

There are various properties associated with transposition. For matrices A and B, given as,

- (A T ) T = A
- (A + B) T = A T + B T , A and B being of the same order.
- (KA) T = KA T , K is any scalar( real or complex ).
- (AB) T = B T A T , A and B being conformable for the product AB. (This is also called reversal law.)

## Trace of a Matrix

## Determinant of Matrices

Then determinant formula of matrix A:

## Minor of Matrix

\(M_{12} = \left|\begin{array}{ccc} a_{21} & a_{23} \\ \\ a_{31} & a_{33} \end{array}\right|\)

## Cofactor of Matrix

\(C_{ij} = (-1)^{i+j} M_{ij}\)

On finding all the cofactors of the matrix, we will get a cofactor matrix C of the given matrix A:

Note: Be extra cautious about the negative sign while calculating the cofactor of the matrix.

## Adjoint of Matrices

Then the minor matrix M of the given matrix would be:

\(M = \left[\begin{array}{ccc} -8 & -2 & -5 \\ 5 & -7 & -1 \\ -17 & 4 & 10 \end{array}\right] \)

We will get the cofactor matrix C of the given matrix A as:

\(C = \left[\begin{array}{ccc} -8 & 2 & -5 \\ -5 & -7 & 1 \\ -17 & -4 & 10 \end{array}\right] \)

Then the transpose of the cofactor matrix will give the adjoint of the given matrix:

## Inverse of Matrices

And on calculating the determinant , we will get |A| = -33

## Types of Matrices

D T = \(\left[\begin{array}{lll} 2 & 3 & 6 \\ 3 & 4 & 5 \\ 6 & 5 & 9 \end{array}\right]\) = D

- F T = \(\left[\begin{array}{cc} 0 & -3\\ \\ 3 & 0 \end{array}\right]\)
- -F = \(\left[\begin{array}{cc} 0 & -3\\ \\ 3 & 0 \end{array}\right]\)

## Solving a System of Equations Using Matrices

\(A = \begin{bmatrix} 1 & 1\\ \\ 2 & 3\\ \end{bmatrix}\)

\(X = \begin{bmatrix} x\\ \\ y\\ \end{bmatrix}\)

\(B = \begin{bmatrix} 8\\ \\ 10\\ \end{bmatrix}\)

\( |A| = \begin{vmatrix} 1 & 1\\ \\ 2 & 3\\ \end{vmatrix}\)

Hence, |A| = 3 - 2 = 1 \(\because\) \(|A| \neq 0\), it is possible to find the inverse of matrix A.

\(A^{-1} = \begin{bmatrix} 3 & -1\\ \\ -2 & 1\\ \end{bmatrix}\)

Now to find the matrix X, we'll multiply \(A^{-1}\) and B. We get,

Hence, the value of matrix X is,

\(X = \begin{bmatrix} 14\\ \\ -6\\ \end{bmatrix}\)

## Rank of a Matrix

## Eigen Values and Eigen Vectors of Matrices

## Matrices Formulas

- A(adj A) = (adj A) A = | A | I n
- | adj A | = | A | n-1
- adj (adj A) = | A | n-2 A
- | adj (adj A) | = | A | (n-1)^2
- adj (AB) = (adj B) (adj A)
- adj (A m ) = (adj A) m ,
- adj (kA) = k n-1 (adj A) , k ∈ R
- adj(I n ) = I n
- A is symmetric ⇒ (adj A) is also symmetric.
- A is diagonal ⇒ (adj A) is also diagonal.
- A is triangular ⇒ adj A is also triangular.
- A is singular ⇒| adj A | = 0
- A -1 = (1/|A|) adj A
- (AB) -1 = B -1 A -1

- Cofactor of the matrix A is obtained when the minor \(M_{ij}\) of the matrix is multiplied with (-1) i+j .
- Matrices are rectangular-shaped arrays.
- The inverse of matrices is calculated by using the given formula: A -1 = (1/|A|)(adj A).
- The inverse of a matrix exists if and only if |A| ≠ 0.

## Solved Examples on Matrices

The given matrix is A = \(\left[\begin{matrix}1 & -2\\ \\2 & -3 \end{matrix}\right]\).

Using the inverse of matrix formula we can calculate A -1 as follows.

A -1 = \(\dfrac{1}{(1× -3) - (-2 × 2)}\left[\begin{matrix}-3&2\\ \\-2&1\end{matrix}\right]\)

= \(\dfrac{1}{-3 +4}\left[\begin{matrix}-3&2\\ \\-2&1\end{matrix}\right]\)

= \(\left[\begin{matrix}-3&2\\ \\-2&1\end{matrix}\right]\)

Answer: Therefore A -1 = \(\left[\begin{matrix}-3&2\\ \\-2&1\end{matrix}\right]\).

= \(\left[\begin{array}{rr}1 & 2 & -1\\ 3 & 2 & 0\\ -4 & 0 & 2\end{array}\right]\)

Answer: We have proved that AI = A.

go to slide go to slide go to slide

## Practice Questions on Matrices

## FAQs on Matrices

What is the meaning of matrix in math.

## How to Solve Matrices?

- For the addition/subtraction of 2 matrices, their orders should be the same.
- For the multiplication of matrices , the number of columns of the left side matrix should be equal to the number of rows of the right side matrix.

## How to Solve Systems of Equations with Matrices?

To solve the system of equations with matrices, we will follow the steps given below.

- Arrange the elements of equations in matrices and find the coefficient matrix, variable matrix, and constant matrix.
- Write the equations in AX = B form.
- Take the inverse of A by finding the adjoint and determinant of A.
- Multiply the inverse of A to matrix B, thereby finding the value of variable matrix X.

## What is 3×3 Inverse Matrix Formula?

## What is the Special Feature Of the Determinant Formula For Matrices?

## How To Calculate the Determinant of a 2×2 Matrix Using Determinant Formula?

## What is the Condition for Matrix Multiplication to be Possible?

## What Are Properties of Transposition of Matrices?

## What is the Formula for Inverse of Matrices?

## How To Use Inverse of Matrix Formula?

The inverse matrix formula can be used following the given steps:

- Step 1: Find the matrix of minors for the given matrix.
- Step 2: Transform the minor matrix so obtained into the matrix of cofactors .
- Step 3: Find the adjoint matrix by taking the transpose of the cofactor matrix.
- Step 4: Finally divide the adjoint of a matrix by its determinant.

## What are the Different Types of a Matrix?

- Row matrix and column matrix
- Square matrix and a rectangular matrix
- Diagonal matrix
- Scalar matrix
- Identity matrix
- Null matrix
- Upper triangular matrix and lower triangular matrix
- Idempotent matrix
- Symmetric and skew-symmetric matrix

## What are the Properties of Scalar Multiplication in Matrices?

## What is a Matrix Polynomial?

## What is the Echelon Form of Matrices?

## How to Express a Matrix as a Sum of Symmetric and Non-Symmetric Matrix?

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## Top 3 Problem-Solving-Based Interview Questions

Alright, here is what you’ve been waiting for: the problem-solving questions and sample answers.

## 1. Can you tell me about a time when you had to solve a challenging problem?

“While working as a mobile telecom support specialist for a large organization, we had to transition our MDM service from one vendor to another within 45 days. This personally physically handling 500 devices within the agency. Devices had to be gathered from the headquarters and satellite offices, which were located all across the state, something that was challenging even without the tight deadline. I approached the situation by identifying the location assignment of all personnel within the organization, enabling me to estimate transit times for receiving the devices. Next, I timed out how many devices I could personally update in a day. Together, this allowed me to create a general timeline. After that, I coordinated with each location, both expressing the urgency of adhering to deadlines and scheduling bulk shipping options. While there were occasional bouts of resistance, I worked with location leaders to calm concerns and facilitate action. While performing all of the updates was daunting, my approach to organizing the event made it a success. Ultimately, the entire transition was finished five days before the deadline, exceeding the expectations of many.”

## 2. Describe a time where you made a mistake. What did you do to fix it?

“When I first began in a supervisory role, I had trouble setting down my individual contributor hat. I tried to keep up with my past duties while also taking on the responsibilities of my new role. As a result, I began rushing and introduced an error into the code of the software my team was updating. The error led to a memory leak. We became aware of the issue when the performance was hindered, though we didn’t immediately know the cause. I dove back into the code, reviewing recent changes, and, ultimately, determined the issue was a mistake on my end. When I made that discovery, I took several steps. First, I let my team know that the error was mine and let them know its nature. Second, I worked with my team to correct the issue, resolving the memory leak. Finally, I took this as a lesson about delegation. I began assigning work to my team more effectively, a move that allowed me to excel as a manager and help them thrive as contributors. It was a crucial learning moment, one that I have valued every day since.”

## 3. If you identify a potential risk in a project, what steps do you take to prevent it?

“If I identify a potential risk in a project, my first step is to assess the various factors that could lead to a poor outcome. Prevention requires analysis. Ensuring I fully understand what can trigger the undesired event creates the right foundation, allowing me to figure out how to reduce the likelihood of those events occurring. Once I have the right level of understanding, I come up with a mitigation plan. Exactly what this includes varies depending on the nature of the issue, though it usually involves various steps and checks designed to monitor the project as it progresses to spot paths that may make the problem more likely to happen. I find this approach effective as it combines knowledge and ongoing vigilance. That way, if the project begins to head into risky territory, I can correct its trajectory.”

## 17 More Problem-Solving-Based Interview Questions

- How would you describe your problem-solving skills?
- Can you tell me about a time when you had to use creativity to deal with an obstacle?
- Describe a time when you discovered an unmet customer need while assisting a customer and found a way to meet it.
- If you were faced with an upset customer, how would you diffuse the situation?
- Tell me about a time when you had to troubleshoot a complex issue.
- Imagine you were overseeing a project and needed a particular item. You have two choices of vendors: one that can deliver on time but would be over budget, and one that’s under budget but would deliver one week later than you need it. How do you figure out which approach to use?
- Your manager wants to upgrade a tool you regularly use for your job and wants your recommendation. How do you formulate one?
- A supplier has said that an item you need for a project isn’t going to be delivered as scheduled, something that would cause your project to fall behind schedule. What do you do to try and keep the timeline on target?
- Can you share an example of a moment where you encountered a unique problem you and your colleagues had never seen before? How did you figure out what to do?
- Imagine you were scheduled to give a presentation with a colleague, and your colleague called in sick right before it was set to begin. What would you do?
- If you are given two urgent tasks from different members of the leadership team, both with the same tight deadline, how do you choose which to tackle first?
- Tell me about a time you and a colleague didn’t see eye-to-eye. How did you decide what to do?
- Describe your troubleshooting process.
- Tell me about a time where there was a problem that you weren’t able to solve. What happened?
- In your opening, what skills or traits make a person an exceptional problem-solver?
- When you face a problem that requires action, do you usually jump in or take a moment to carefully assess the situation?
- When you encounter a new problem you’ve never seen before, what is the first step that you take?

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## Decision Matrix Resources

## Decision Matrix Related Topics

## What is a Decision Matrix?

Quality Glossary Definition: Decision matrix

## When to Use a Decision Matrix

- When a list of options must be narrowed to one choice
- When the decision must be made on the basis of several criteria
- After a list of options has been reduced to a manageable number by list reduction

- When one improvement opportunity or problem must be selected to work on
- When only one solution or problem-solving approach can be implemented
- When only one new product can be developed

## Decision Matrix Procedure

- Brainstorm the evaluation criteria appropriate to the situation. If possible, involve customers in this process.
- Discuss and refine the list of criteria. Identify any criteria that must be included and any that must not be included. Reduce the list of criteria to those that the team believes are most important. Tools such as list reduction and multivoting may be useful.
- By distributing 10 points among the criteria, based on team discussion and consensus.
- By each member assigning weights, then the numbers for each criterion for a composite team weighting.
- Draw an L-shaped matrix. Write the criteria and their weights as labels along one edge and the list of options along the other edge. Typically, the group with fewer items occupies the vertical edge.

Method 1: Establish a rating scale for each criterion. Some options are:

- 1, 2, 3 (1 = slight extent, 2 = some extent, 3 = great extent)
- 1, 2, 3 (1 = low, 2 = medium, 3 = high)
- 1, 2, 3, 4, 5 (1 = little to 5 = great)
- 1, 4, 9 (1 = low, 4 = moderate, 9 = high)

## Decision Matrix Example

Figure 1: Decision Matrix Example

## Decision Matrix Considerations

- A very long list of options can first be shortened with a tool such as list reduction or multivoting .
- Within control of the team
- Financial payback
- Resources required (e.g., money, people)
- Customer pain caused by the problem
- Urgency of problem
- Team interest or buy-in
- Effect on other systems
- Management interest or support
- Difficulty of solving
- Time required to solve
- Root causes addressed by this solution
- Extent of resolution of problem
- Cost to implement ( e.g., money, time)
- Return on investment; availability of resources ( e.g., people, time)
- Ease of implementation
- Time until solution is fully implemented
- Cost to maintain ( e.g., money, time)
- Ease of maintenance
- Support or opposition to the solution
- Enthusiasm by team members
- Team control of the solution
- Safety, health, or environmental factors
- Training factors
- Potential effects on other systems
- Potential effects on customers or suppliers
- Value to customer
- Potential problems during implementation
- Potential negative consequences

## Additional considerations

- While a decision matrix can be used to compare opinions, it is better used to summarize data that have been collected about the various criteria when possible.
- Sub-teams can be formed to collect data on the various criteria.
- Several criteria for selecting a problem or improvement opportunity require guesses about the ultimate solution. For example: evaluating resources required, payback, difficulty to solve, and time required to solve. Therefore, your rating of the options will be only as good as your assumptions about the solutions.
- If individuals on the team assign different ratings to the same criterion, discuss until the team arrives at a consensus. Do not average the ratings or vote for the most popular one.
- In some versions of this tool, the sum of the unweighted scores is also calculated and both totals are studied for guidance toward a decision.
- When this tool is used to choose a plan, solution, or new product, results can be used to improve options. An option that ranks highly overall but has low scores on criteria A and B can be modified with ideas from options that score well on A and B. This combining and improving can be done for every option, and then the decision matrix used again to evaluate the new options.

Adapted from The Quality Toolbox, Second Edition, ASQ Quality Press.

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Precalculus Unit 7: Lesson 10 Multiplying matrices by matrices Intro to matrix multiplication Multiplying matrices Multiplying matrices Multiply matrices Math > Precalculus > Matrices > Multiplying matrices by matrices Multiply matrices CCSS.Math: HSN.VM.C.8 Google Classroom

Matrix word problem: prices About this unit This topic covers: - Adding & subtracting matrices - Multiplying matrices by scalars - Multiplying matrices - Representing & solving linear systems with matrices - Matrix inverses - Matrix determinants - Matrices as transformations - Matrices applications

Problem 5. A square matrix Aover C is called skew-hermitian if A= A. Show that such a matrix is normal, i.e., we have AA = AA. Problem 6. Let Abe an n nskew-hermitian matrix over C, i.e. A = A. Let U be an n n unitary matrix, i.e., U = U 1. Show that B:= U AUis a skew-hermitian matrix. Problem 7. Let A, X, Y be n nmatrices. Assume that XA= I n ...

Questions on Matrices: Part A Given the matrices: a) What is the dimension of each matrix? b) Which matrices are square? c) Which matrices are symmetric? d) Which matrix has the entry at row 3 and column 2 equal to -11? e) Which matrices has the entry at row 1 and column 3 equal to 10? f) Which are column matrices? g) Which are row matrices?

Matrices and Determinants: Problems with Solutions Matrices Matrix multiplication Determinants Rank of matrices Inverse matrices Matrix equations Systems of equations Matrix calculators Problem 1 What are the dimensions of the matrix \displaystyle A A?

The Matrix Solution. Then (also shown on the Inverse of a Matrix page) the solution is this: X = BA -1. This is what we get for A-1: In fact it is just like the Inverse we got before, but Transposed (rows and columns swapped over). Next we multiply B by A-1: And the solution is the same: x = 5, y = 3 and z = −2.

Solved Example Problems on Applications of Matrices: Solving System of Linear Equations Solution to a System of Linear equations: (i) Matrix Inversion Method (ii) Cramer's Rule (iii) Gaussian Elimination Method Solution to a System of Linear equations (i) Matrix Inversion Method Example 1.22

Practice Questions on Matrix Multiplication 1. Find the product of the following matrices: A = [ 1 − 2 3 3 2 − 1] a n d B = [ 2 3 − 1 2 4 − 5] 2. Let X, Y, Z, W and S are matrices of order 2 × n, 3 × k, 2 × p, n × 3 and p × k, respectively. Find the order of matrix 7X - 5Z, if n = p. 3.

Step 6 : Get prepared to write down your objective data on your Matrix Solver Forms. Your objective data includes your: Key Questions, Information (answers), Tasks, and Goals. You'll be grouping them by: Who, What, When, Where, Why, How, From Where, and To Where. Step 7 : Start with your Matrix Solver Form for 'Who'.

Examples Frequently Asked Questions (FAQ) How do you multiply two matrices together? To multiply two matrices together the inner dimensions of the matrices shoud match.

The solution matrix will look like this: [6] 1 0 0 x 0 1 0 y 0 0 1 z Notice that the matrix consists of 1's in a diagonal line with 0's in all other spaces, except the fourth column. The numbers in the fourth column will be your solution for the variables x, y and z. 2 Use scalar multiplication.

A matrix is a rectangular array of numbers, variables, symbols, or expressions that are defined for the operations like subtraction, addition, and multiplications. The size of a matrix (which is known as the order of the matrix) is determined by the number of rows and columns in the matrix.The order of a matrix with 6 rows and 4 columns is represented as a 6 × 4 and is read as 6 by 4.

The word "process.". In the end, problem-solving is an activity. It's your ability to take appropriate steps to find answers, determine how to proceed, or otherwise overcome the challenge. Being great at it usually means having a range of helpful problem-solving skills and traits. Research, diligence, patience, attention-to-detail ...

a more general problem (this is the kind of thing mathematicians love to do) in which we do not know exactly what the coeﬃcients are (ie: 1, 2/3, 1/2, 1800, 1100): ... understand how they help to solve linear equations. 3 Matrices and matrix multiplication A matrix is any rectangular array of numbers. If the array has n rows and m columns ...

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Quality Glossary Definition: Decision matrix. Also called: Pugh matrix, decision grid, selection matrix or grid, problem matrix, problem selection matrix, opportunity analysis, solution matrix, criteria rating form, criteria-based matrix. A decision matrix evaluates and prioritizes a list of options and is a decision-making tool.

You might be also interested in: - Sum, Difference and Product of Matrices. - Inverse Matrix. - Rank of a Matrix. - Determinant of a Matrix. - System of Equations Solved by Matrices. - Matrix Word Problems. Link Partners.

Matrix Calculator. Type a math problem. Type a math problem. Solve ... Get step-by-step explanations. See how to solve problems and show your work—plus get definitions for mathematical concepts. Graph your math problems. Instantly graph any equation to visualize your function and understand the relationship between variables. Practice ...

Matrix math exercises & matrices math problems for students of all ages. Matrix equations. Math-Exercises.com - Math exercises with correct answers.

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