What is the order of the product of a 2×3 and 3×4 matrix?

The resulting matrix will be of order 2×4.

Can we multiply a 2×2 matrix with a 2×3 matrix?

Yes, if the number of columns of the first equals the number of rows of the second (2×2) × (2×3) is not valid. But (2×3) × (3×2) is valid.

What is the identity matrix?

An identity matrix is a square matrix with 1s on the diagonal and 0s elsewhere. Multiplying any matrix with the identity matrix gives the original matrix.

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Matrix Multiplication Questions

Matrix multiplication is a fundamental concept in linear algebra and plays a key role in various applications like computer graphics, engineering, and physics. In competitive exams like JEE Main and Advanced, matrix multiplication questions test your understanding of matrix operations, properties, and computation techniques. This blog presents a comprehensive set of matrix multiplication questions and answers, including basic to advanced-level problems, step-by-step solutions, and practice exercises to strengthen your problem-solving skills.

1.0What is Matrix Multiplication?

Matrix multiplication is an operation where two matrices are multiplied to produce a third matrix. If matrix A is of order (m × n) and matrix B is of order (n × p), their product AB will be a matrix of order (m × p).

Conditions for Matrix Multiplication

The number of columns in the first matrix must equal the number of rows in the second matrix.

2.0Matrix Multiplication Questions and Answers

Example 1: Multiply the matrices: $A = [1324], B = [5768]$

Solution:

$A = [1324], B = [5768]$

$A B = A = [(1 \times 5 + 2 \times 7) (3 \times 5 + 4 \times 7) (1 \times 6 + 2 \times 8) (3 \times 6 + 4 \times 8)] = [19432250]$

Example 2: If $A = [42 - 3 1]$ , find AI, where I is the identity matrix of order 2.

Solution:

$I = [1001], A I = A$

$A I = [42 - 3 1]$

Multiplying any matrix with an identity matrix (of compatible order) gives the matrix itself.

Example 3: Find the product of matrices: $A = [2, 5, 3], B = 41 - 2$

Solution:

$A B = (2 \times 4) + (5 \times 1) + (3 \times (- 2))$

$A B = 8 + 5 - 6 = 7$

So, the product is: $A B = [7]$

Example 4: Is matrix multiplication commutative?

Let $A = [1021], B = [3214]$ . Find AB and BA.

Solution:

$A = [1021], B = [3214]$

$A B = [(1 \times 3 + 2 \times 2) (0 \times 3 + 1 \times 2) (1 \times 1 + 2 \times 4) (0 \times 1 + 1 \times 4)] = [7294]$

$B A = [(3 \times 1 + 1 \times 0) (2 \times 1 + 4 \times 0) (3 \times 2 + 1 \times 1) (2 \times 2 + 4 \times 1)] = [3278]$

Clearly, AB ≠ BA.

Matrix multiplication is not commutative.

Example 5: Let $A = [21 - 1 0 3 - 2]$ . Find $A A^{T}$ .

Solution:

Step 1: First, compute the transpose: $A^{T} 2 - 1 3 10 - 2$

Step 2: Multiply A and $A^{T}$ :

$A A^{T} = [21 - 1 0 3 - 2] 2 - 1 3 10 - 2$

$= [(2 \times) 2 + (- 1) \times (- 1) + 3 \times 3 (1 \times 2 + 0 \times (- 1) + (- 2) \times 3) (2 \times 1 + (- 1) \times 0 + 3 \times (- 2)) (1 \times 1 + 0 \times 0 + (- 2) \times (- 2))]$

$= [4 + 1 + 9 2 + 0 - 6 2 + 0 - 6 1 + 0 + 4] = [14 - 4 - 4 5]$

Example 6: Let $A = [1021], B = [x z y w]$ . Find the condition(s) on x, y, z, w such that AB = BA.

Solution:

Compute both:

$A B = [1 \times x + 2 \times z 0 \times x + 1 \times z 1 \times y + 2 \times w 0 \times y + 1 \times w] = [x + 2 z z y + 2 w w]$

$B A = [x \times 1 + y \times 0 z \times 1 + w \times 0 x \times 2 + y \times 1 z \times 2 + w \times 1] = [x z 2 x + y 2 z + w]$

Equating AB = BA:

$x + 2 z = x, y + 2 w = 2 x + y, z = z, w = 2 z + w$

From equations:

$x + 2 z = x \Rightarrow z = 0$

$y + 2 w = 2 x + y$

$\Rightarrow 2 w = 2 x$

$\Rightarrow w = x$

Now,

$w = 2 z + w$

$\Rightarrow 0 = 2 z$

$\Rightarrow z = 0 (already satisfied)$

Hence, z = 0 and w = x are required.

3.0Practice Matrix Multiplication Questions

Question 1: If $A = [1 - 1 02], B = [3041]$ . Find AB and BA.

Question 2: Find the product of $A = [20 - 1 3], B = [4501]$

Question 3: If $A = [a c b d], B = [1000]$ . Then what is AB?

Question 4: Verify whether matrix multiplication is associative for the following matrices: $A = [12], B = [01], C = [3]$ . Check if A(BC) = (AB)C

Also Practice : Matrices and Determinants previous year questions with solutions

4.0Tips for Solving Matrix Multiplication Questions

Always check matrix order before attempting multiplication.
Use brackets and align terms while calculating.
Practice identity and zero matrix properties.
Remember AB ≠ BA in most cases.
Know special cases like row × column and scalar multiplication.

Also Explore:

Elimination Method	Find Inverse of Matrix	Rank of Matrix
Adjoint of a Matrix	Transpose of a Matrix	Adjacency Matrix
Types of Matrices	Application of Matrices	Determinant of a Matrix

Join ALLEN!

(Session 2026 - 27)

Name

Mobile Number

Class

Choose class

Goal

Choose your goal

Preferred Programs

Preferred Mode

State

Choose State

I agree to

I authorise ALLEN Career Institute Pvt Ltd to send me regular updates via Phone calls, Whatsapp, SMS, Robocalls (Automated Calls), Emails, or on postal address.

Matrix Multiplication Questions

Matrix multiplication is a fundamental concept in linear algebra and plays a key role in various applications like computer graphics, engineering, and physics. In competitive exams like JEE Main and Advanced, matrix multiplication questions test your understanding of matrix operations, properties, and computation techniques. This blog presents a comprehensive set of matrix multiplication questions and answers, including basic to advanced-level problems, step-by-step solutions, and practice exercises to strengthen your problem-solving skills.

1.0What is Matrix Multiplication?

Matrix multiplication is an operation where two matrices are multiplied to produce a third matrix. If matrix A is of order (m × n) and matrix B is of order (n × p), their product AB will be a matrix of order (m × p).

Conditions for Matrix Multiplication

The number of columns in the first matrix must equal the number of rows in the second matrix.

2.0Matrix Multiplication Questions and Answers

Example 1: Multiply the matrices: $A = [1324], B = [5768]$

Solution:

$A = [1324], B = [5768]$

$A B = A = [(1 \times 5 + 2 \times 7) (3 \times 5 + 4 \times 7) (1 \times 6 + 2 \times 8) (3 \times 6 + 4 \times 8)] = [19432250]$

Example 2: If $A = [42 - 3 1]$ , find AI, where I is the identity matrix of order 2.

Solution:

$I = [1001], A I = A$

$A I = [42 - 3 1]$

Multiplying any matrix with an identity matrix (of compatible order) gives the matrix itself.

Example 3: Find the product of matrices: $A = [2, 5, 3], B = 41 - 2$

Solution:

$A B = (2 \times 4) + (5 \times 1) + (3 \times (- 2))$

$A B = 8 + 5 - 6 = 7$

So, the product is: $A B = [7]$

Example 4: Is matrix multiplication commutative?

Let $A = [1021], B = [3214]$ . Find AB and BA.

Solution:

$A = [1021], B = [3214]$

$A B = [(1 \times 3 + 2 \times 2) (0 \times 3 + 1 \times 2) (1 \times 1 + 2 \times 4) (0 \times 1 + 1 \times 4)] = [7294]$

$B A = [(3 \times 1 + 1 \times 0) (2 \times 1 + 4 \times 0) (3 \times 2 + 1 \times 1) (2 \times 2 + 4 \times 1)] = [3278]$

Clearly, AB ≠ BA.

Matrix multiplication is not commutative.

Example 5: Let $A = [21 - 1 0 3 - 2]$ . Find $A A^{T}$ .

Solution:

Step 1: First, compute the transpose: $A^{T} 2 - 1 3 10 - 2$

Step 2: Multiply A and $A^{T}$ :

$A A^{T} = [21 - 1 0 3 - 2] 2 - 1 3 10 - 2$

$= [(2 \times) 2 + (- 1) \times (- 1) + 3 \times 3 (1 \times 2 + 0 \times (- 1) + (- 2) \times 3) (2 \times 1 + (- 1) \times 0 + 3 \times (- 2)) (1 \times 1 + 0 \times 0 + (- 2) \times (- 2))]$

$= [4 + 1 + 9 2 + 0 - 6 2 + 0 - 6 1 + 0 + 4] = [14 - 4 - 4 5]$

Example 6: Let $A = [1021], B = [x z y w]$ . Find the condition(s) on x, y, z, w such that AB = BA.

Solution:

Compute both:

$A B = [1 \times x + 2 \times z 0 \times x + 1 \times z 1 \times y + 2 \times w 0 \times y + 1 \times w] = [x + 2 z z y + 2 w w]$

$B A = [x \times 1 + y \times 0 z \times 1 + w \times 0 x \times 2 + y \times 1 z \times 2 + w \times 1] = [x z 2 x + y 2 z + w]$

Equating AB = BA:

$x + 2 z = x, y + 2 w = 2 x + y, z = z, w = 2 z + w$

From equations:

$x + 2 z = x \Rightarrow z = 0$

$y + 2 w = 2 x + y$

$\Rightarrow 2 w = 2 x$

$\Rightarrow w = x$

Now,

$w = 2 z + w$

$\Rightarrow 0 = 2 z$

$\Rightarrow z = 0 (already satisfied)$

Hence, z = 0 and w = x are required.

3.0Practice Matrix Multiplication Questions

Question 1: If $A = [1 - 1 02], B = [3041]$ . Find AB and BA.

Question 2: Find the product of $A = [20 - 1 3], B = [4501]$

Question 3: If $A = [a c b d], B = [1000]$ . Then what is AB?

Question 4: Verify whether matrix multiplication is associative for the following matrices: $A = [12], B = [01], C = [3]$ . Check if A(BC) = (AB)C

Also Practice : Matrices and Determinants previous year questions with solutions

4.0Tips for Solving Matrix Multiplication Questions

Always check matrix order before attempting multiplication.
Use brackets and align terms while calculating.
Practice identity and zero matrix properties.
Remember AB ≠ BA in most cases.
Know special cases like row × column and scalar multiplication.

Also Explore:

Elimination Method	Find Inverse of Matrix	Rank of Matrix
Adjoint of a Matrix	Transpose of a Matrix	Adjacency Matrix
Types of Matrices	Application of Matrices	Determinant of a Matrix

Frequently Asked Questions

What is the order of the product of a 2×3 and 3×4 matrix?

Can we multiply a 2×2 matrix with a 2×3 matrix?

What is the identity matrix?

Frequently Asked Questions

What is the order of the product of a 2×3 and 3×4 matrix?

Can we multiply a 2×2 matrix with a 2×3 matrix?

What is the identity matrix?

Join ALLEN!

Matrix Multiplication Questions

1.0What is Matrix Multiplication?

Conditions for Matrix Multiplication

2.0Matrix Multiplication Questions and Answers

3.0Practice Matrix Multiplication Questions

4.0Tips for Solving Matrix Multiplication Questions

Table of Contents

Join ALLEN!

Matrix Multiplication Questions

1.0What is Matrix Multiplication?

Conditions for Matrix Multiplication

2.0Matrix Multiplication Questions and Answers

3.0Practice Matrix Multiplication Questions

4.0Tips for Solving Matrix Multiplication Questions

Table of Contents