If $A$ is a $3 \times 3$ matrix and $B$ is a $3 \times 3$ matrix, under what condition is the product $AB$ equal to $BA$?

If $A$ and $B$ commute, i.e., $AB = BA$.

If $A$ and $B$ are both diagonal matrices.

If $A$ and $B$ are both identity matrices.

If $A$ and $B$ are both zero matrices.

If $A$ and $B$ commute, i.e., $AB = BA$.

Matrices

Action Plan Notes Questions Mock-Test

Study Action Plan (Sorted by Difficulty)

Inverse Matrix Properties

Accuracy: 43%

Total Attempted23

Total Correct10

Total Incorrect13

Difficulty Level:

High

Zero Product Property Failure in Matrices

Accuracy: 44%

Total Attempted16

Total Correct7

Total Incorrect9

Difficulty Level:

High

Properties of Matrix Multiplication

Accuracy: 47%

Total Attempted15

Total Correct7

Total Incorrect8

Difficulty Level:

High

Matrix Multiplication

Accuracy: 48%

Total Attempted60

Total Correct29

Total Incorrect31

Difficulty Level:

High

Non-Commutativity of Matrix Multiplication

Accuracy: 50%

Total Attempted10

Total Correct5

Total Incorrect5

Difficulty Level:

Medium

Properties of Matrix Operations

Accuracy: 54%

Total Attempted39

Total Correct21

Total Incorrect18

Difficulty Level:

Medium

Properties of Matrix Addition and Scalar Multiplication

Accuracy: 54%

Total Attempted57

Total Correct31

Total Incorrect26

Difficulty Level:

Medium

Conditions and Order of Matrix Products

Accuracy: 56%

Total Attempted71

Total Correct40

Total Incorrect31

Difficulty Level:

Medium

Transpose Properties and Product Rule

Accuracy: 57%

Total Attempted14

Total Correct8

Total Incorrect6

Difficulty Level:

Medium

Matrix Equations and Systems

Accuracy: 59%

Total Attempted37

Total Correct22

Total Incorrect15

Difficulty Level:

Medium

Symmetric and Skew-Symmetric Matrices

Accuracy: 60%

Total Attempted80

Total Correct48

Total Incorrect32

Difficulty Level:

Medium

Applications of Matrix Multiplication

Accuracy: 62%

Total Attempted13

Total Correct8

Total Incorrect5

Difficulty Level:

Medium

Matrix Algebra

Accuracy: 68%

Total Attempted107

Total Correct73

Total Incorrect34

Difficulty Level:

Medium

Types of Matrices

Accuracy: 69%

Total Attempted59

Total Correct41

Total Incorrect18

Difficulty Level:

Medium

Invertible Matrices

Accuracy: 74%

Total Attempted61

Total Correct45

Total Incorrect16

Difficulty Level:

Medium

Equality of Matrices

Accuracy: 76%

Total Attempted50

Total Correct38

Total Incorrect12

Difficulty Level:

Low

Transpose of a Matrix

Accuracy: 80%

Total Attempted66

Total Correct53

Total Incorrect13

Difficulty Level:

Low

Construction of Matrices from General Term

Accuracy: 83%

Total Attempted42

Total Correct35

Total Incorrect7

Difficulty Level:

Low

Scalar Multiplication and Negative of Matrix

Accuracy: 90%

Total Attempted69

Total Correct62

Total Incorrect7

Difficulty Level:

Low

Matrix Definition, Elements and Order

Accuracy: 91%

Total Attempted57

Total Correct52

Total Incorrect5

Difficulty Level:

Low

Addition and Subtraction of Matrices

Accuracy: 93%

Total Attempted43

Total Correct40

Total Incorrect3

Difficulty Level:

Low

Decomposition into Symmetric and Skew-Symmetric Parts

Accuracy: 100%

Total Attempted17

Total Correct17

Total Incorrect0

Difficulty Level:

Low

Learning Objectives

Top

Understand the concept and definition of matrices.
Identify the various types of matrices, including symmetric and skew symmetric matrices.
Perform matrix operations such as addition, subtraction, and multiplication.
Apply the properties of matrix operations, including commutative, associative, and distributive laws.
Use matrices to solve systems of linear equations.
Explore the applications of matrices in different fields such as business, science, and cryptography.
Verify matrix identities using examples, such as $(AB)' = B'A'$ .
Calculate the inverse of a matrix and understand its uniqueness.
Express any square matrix as the sum of a symmetric and a skew symmetric matrix.
Prove properties of matrices using mathematical induction and other methods.

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Revision Notes & Summary

Top

Matrices

Introduction

Matrices are essential in various branches of mathematics.
They simplify solving systems of linear equations and are used in electronic spreadsheets for business and science applications.
Matrices represent physical operations like magnification, rotation, and reflection, and are used in cryptography, genetics, economics, sociology, psychology, and industrial management.

Matrix Basics

A matrix is an ordered rectangular array of numbers or functions.
A matrix with $m$ rows and $n$ columns is of order $m \times n$ .
Types of matrices:
- Column Matrix: $[a_{ij}]_{m \times 1}$
- Row Matrix: $[a_{ij}]_{1 \times n}$
- Square Matrix: When $m = n$ .
- Diagonal Matrix: $a_{ij} = 0$ when $i \neq j$ .
- Scalar Matrix: $a_{ij} = 0$ when $i \neq j$ , $a_{ij} = k$ when $i = j$ .
- Identity Matrix: $a_{ij} = 1$ when $i = j$ , $a_{ij} = 0$ when $i \neq j$ .
- Zero Matrix: All elements are zero.

Matrix Operations

Addition: $A + B = B + A$
Multiplication: If $A = [a_{ij}]_{m \times n}$ and $B = [b_{jk}]_{n \times p}$ , then $AB = C = [c_{ik}]_{m \times p}$ .
Transpose: $A' = [a_{ji}]_{n \times m}$ $A^{'} = [a_{ji}]_{n \times m}$
- Properties:
  - $(A')' = A$
  - $(kA)' = kA'$
  - $(A + B)' = A' + B'$
  - $(AB)' = B'A'$

Special Matrices

Symmetric Matrix: $A' = A$
Skew Symmetric Matrix: $A' = -A$
Any square matrix can be represented as the sum of a symmetric and a skew symmetric matrix.

Invertible Matrices

A square matrix $A$ is invertible if there exists a matrix $B$ such that $AB = BA = I$ .
The inverse of a matrix, if it exists, is unique.
If $A$ and $B$ are invertible, then $(AB)^{-1} = B^{-1}A^{-1}$ .

Examples

Example 1: If $A = \begin{bmatrix} 2 & 3 \\ 1 & 4 \end{bmatrix}$ , find $A'$ , $A^{-1}$ , and verify $AA^{-1} = I$ .
Example 2: Given matrices $A$ and $B$ , show $(A + B)' = A' + B'$ .

Applications

Used in solving linear equations, business applications, physical transformations, and various scientific fields.

Exercises

Prove properties of transpose and inverse matrices.
Solve matrix equations and verify results using matrix operations.

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Practice Test – MCQs, True/False

Top

Multiple Choice Questions

A and B are equal matrices.

A and B are not equal matrices.

A and B have different orders.

A is a scalar multiple of B.

Correct Answer: A

Solution:

Since all corresponding elements are equal and they have the same order, matrices A and B are equal.

Chapter Concept:

Equality of Matrices

A = \frac{1}{2}(A + A') + \frac{1}{2}(A - A')

A = A' + A

A = A - A'

A = A + A'

Correct Answer: A

Solution:

The decomposition of a square matrix

A

into symmetric and skew-symmetric parts is given by

A = \frac{1}{2}(A + A') + \frac{1}{2}(A - A')

, where the first term is symmetric and the second term is skew-symmetric.

Chapter Concept:

Decomposition into Symmetric and Skew-Symmetric Parts

AB is symmetric.

BA is symmetric.

AB = BA.

AB is skew-symmetric.

Correct Answer: B

Solution:

Calculating both products,

BA

is symmetric as it equals

\begin{bmatrix} 3 & 1 \\ 4 & 2 \end{bmatrix}

, which is symmetric.

Chapter Concept:

Matrix Algebra

A

and

B

are both diagonal matrices.

A

and

B

are both identity matrices.

A

and

B

are both zero matrices.

A

and

B

commute, i.e.,

AB = BA

Correct Answer: D

Solution:

For two matrices

A

and

B

to commute,

AB

must equal

BA

. This is not generally true for all matrices unless specified by certain conditions such as both being diagonal matrices.

Chapter Concept:

Conditions and Order of Matrix Products

B

is the inverse of

A

B

is a diagonal matrix

B

is a zero matrix

B

is a row matrix

Correct Answer: A

Solution:

Matrix

B

is the inverse of

A

AB = BA = I

, where

I

is the identity matrix.

Chapter Concept:

Invertible Matrices

(AB)' = A'B'

(AB)' = B'A'

(AB)' = A' + B'

(AB)' = B' + A'

Correct Answer: B

Solution:

The transpose of a product of two matrices

A

and

B

is given by

(AB)' = B'A'

, which is a fundamental property of matrix transposition.

Chapter Concept:

Transpose Properties and Product Rule

All elements are zero.

All non-diagonal elements are zero.

All diagonal elements are equal.

All diagonal elements are one.

Correct Answer: B

Solution:

A diagonal matrix has all non-diagonal elements as zero.

Chapter Concept:

Types of Matrices

A and B are equal matrices.

A and B are not equal matrices.

A and B have the same determinant.

A is a diagonal matrix.

Correct Answer: B

Solution:

Matrices A and B are not equal because their elements at position (2,2) are different.

Chapter Concept:

Equality of Matrices

When

A

and

B

commute, i.e.,

AB = BA

When both

A

and

B

are diagonal matrices.

When both

A

and

B

are zero matrices.

When

A

is the identity matrix.

Correct Answer: A

Solution:

For

AB

to be symmetric, it must be that

(AB)' = AB

. Since

(AB)' = B'A'

and both

A

and

B

are symmetric, this implies

BA = AB

Chapter Concept:

Properties of Matrix Multiplication

\begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}

\begin{bmatrix} 2 & 3 \\ -1 & 4 \end{bmatrix}

\begin{bmatrix} 3 & 2 \\ 4 & -1 \end{bmatrix}

\begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}

Correct Answer: A

Solution:

The transpose of a matrix

A

is obtained by interchanging its rows and columns. Therefore, the transpose of

A = \begin{bmatrix} 2 & 3 \\ -1 & 4 \end{bmatrix}

A' = \begin{bmatrix} 2 & -1 \\ 3 & 4 \end{bmatrix}

Chapter Concept:

Construction of Matrices from General Term

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Matrices

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True or False

Correct Answer: True

Solution:

Matrix addition is defined as the element-wise addition of two matrices of the same order. If the matrices are not of the same order, addition is not defined.

Chapter Concept :

Addition and Subtraction of Matrices

Correct Answer: True

Solution:

The negative of a matrix is defined as

-A = (-1)A

, where each element of the matrix

A

is multiplied by

-1

. This operation is used in matrix subtraction and to find the additive inverse.

Chapter Concept :

Scalar Multiplication and Negative of Matrix

Correct Answer: False

Solution:

Matrix multiplication is not commutative. Even if

AB

and

BA

are both defined, it is not necessary that

AB = BA

. This is a fundamental property of matrices.

Chapter Concept :

Properties of Matrix Operations

Correct Answer: False

Solution:

Matrix multiplication is not commutative. Even if

AB

and

BA

are both defined, it is not necessary that

AB = BA

. This is a fundamental property of matrix multiplication.

Chapter Concept :

Properties of Matrix Operations

Correct Answer: True

Solution:

Matrix addition is indeed commutative and associative for matrices of the same order, meaning that for any matrices

A

and

B

of the same order,

A + B = B + A

and

(A + B) + C = A + (B + C)

. This property is fundamental to matrix operations.

Chapter Concept :

Properties of Matrix Addition and Scalar Multiplication

Correct Answer: True

Solution:

By definition, a square matrix

A

is symmetric if

A' = A

, meaning the transpose of the matrix is equal to the matrix itself.

Chapter Concept :

Symmetric and Skew-Symmetric Matrices

Correct Answer: True

Solution:

The property

(AB)' = B'A'

holds for any matrices

A

and

B

of suitable orders, as stated in the properties of transpose of matrices.

Chapter Concept :

Transpose of a Matrix

Correct Answer: False

Solution:

The product

AB

is defined and has order

m \times p

, but

BA

is not necessarily defined unless

m = p

. Therefore, both

AB

and

BA

are defined only under specific order conditions.

Chapter Concept :

Conditions and Order of Matrix Products

Correct Answer: True

Solution:

The inverse of a square matrix, if it exists, is unique. This means there is only one matrix

B

such that

AB = BA = I

Chapter Concept :

Invertible Matrices

Correct Answer: True

Solution:

A matrix

A

is symmetric if

A' = A

and skew symmetric if

A' = -A

. If

A

is both symmetric and skew symmetric, then

A = -A

, which implies

2A = 0

, therefore

A

must be the zero matrix.

Chapter Concept :

Matrix Algebra

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Matrices

Study Action Plan (Sorted by Difficulty)

Inverse Matrix Properties

High

Zero Product Property Failure in Matrices

High

Properties of Matrix Multiplication

High

Matrix Multiplication

High

Non-Commutativity of Matrix Multiplication

Medium

Properties of Matrix Operations

Medium

Properties of Matrix Addition and Scalar Multiplication

Medium

Conditions and Order of Matrix Products

Medium

Transpose Properties and Product Rule

Medium

Matrix Equations and Systems

Medium

Symmetric and Skew-Symmetric Matrices

Medium

Applications of Matrix Multiplication

Medium

Matrix Algebra

Medium

Types of Matrices

Medium

Invertible Matrices

Medium

Equality of Matrices

Low

Transpose of a Matrix

Low

Construction of Matrices from General Term

Low

Scalar Multiplication and Negative of Matrix

Low

Matrix Definition, Elements and Order

Low

Addition and Subtraction of Matrices

Low

Decomposition into Symmetric and Skew-Symmetric Parts

Low

Learning Objectives

Practice, Analyze & Improve

Revision Notes & Summary

Matrices

Introduction

Matrix Basics

Matrix Operations

Special Matrices

Invertible Matrices

Examples

Applications

Exercises

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Practice Test – MCQs, True/False

Multiple Choice Questions

Q1.Given matrices A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}A=[13​24​] and B=[1234]B = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}B=[13​24​], which of the following statements is true about AAA and BBB?

Correct Answer: A

Q2.Any square matrix AAA can be decomposed into symmetric and skew-symmetric parts. Which of the following expressions correctly represents this decomposition?

Correct Answer: A

Q3.If A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}A=[13​24​] and B=[0110]B = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}B=[01​10​], which of the following statements is true?

Correct Answer: B

Q4.If AAA is a 3×33 \times 33×3 matrix and BBB is a 3×33 \times 33×3 matrix, under what condition is the product ABABAB equal to BABABA?

Correct Answer: D

Q5.If AAA is a square matrix of order mmm, and there exists another square matrix BBB of the same order such that AB=BA=IAB = BA = IAB=BA=I, what can be said about matrix BBB?

Correct Answer: A

Q6.Which of the following properties is true for the transpose of a product of two matrices AAA and BBB?

Correct Answer: B

Q7.Which of the following is a characteristic of a diagonal matrix?

Correct Answer: B

Q8.If A=[1234]A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}A=[13​24​] and B=[1235]B = \begin{bmatrix} 1 & 2 \\ 3 & 5 \end{bmatrix}B=[13​25​], which statement is correct?

Correct Answer: B

Q9.If AAA and BBB are symmetric matrices of the same order, under what condition is the product ABABAB symmetric?

Correct Answer: A

Q10.Given a matrix A=[23−14]A = \begin{bmatrix} 2 & 3 \\ -1 & 4 \end{bmatrix}A=[2−1​34​], which of the following matrices represents the transpose of AAA?