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: $A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$

Solution:

$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$

$AB=A=\left[\begin{array}{cc}(1\times 5+2\times 7)& (1\times 6+2\times 8)\\ (3\times 5+4\times 7)& (3\times 6+4\times 8)\end{array}\right]=\left[\begin{array}{cc}19& 22\\ 43& 50\end{array}\right]$

Example 2: If $A=\left[\begin{array}{cc}4& -3\\ 2& 1\end{array}\right]$, find AI, where I is the identity matrix of order 2.

Solution:

$I=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right],AI=A$

$AI=\left[\begin{array}{cc}4& -3\\ 2& 1\end{array}\right]$ 

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

Example 3: Find the product of matrices: $A=[2,5,3],B=\left[\begin{array}{c}4\\ 1\\ -2\end{array}\right]$

Solution:

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

$AB=8+5-6=7$

So, the product is: $AB=[7]$

Example 4: Is matrix multiplication commutative?

Let $A=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}3& 1\\ 2& 4\end{array}\right]$. Find AB and BA.

Solution:

$A=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}3& 1\\ 2& 4\end{array}\right]$

$AB=\left[\begin{array}{cc}(1\times 3+2\times 2)& (1\times 1+2\times 4)\\ (0\times 3+1\times 2)& (0\times 1+1\times 4)\end{array}\right]=\left[\begin{array}{cc}7& 9\\ 2& 4\end{array}\right]$

$BA=\left[\begin{array}{cc}(3\times 1+1\times 0)& (3\times 2+1\times 1)\\ (2\times 1+4\times 0)& (2\times 2+4\times 1)\end{array}\right]=\left[\begin{array}{cc}3& 7\\ 2& 8\end{array}\right]$

Clearly, AB ≠ BA.

Matrix multiplication is not commutative.

Example 5: Let $A=\left[\begin{array}{ccc}2& -1& 3\\ 1& 0& -2\end{array}\right]$. Find $A{A}^T$.

Solution:

Step 1: First, compute the transpose: $A^T\left[\begin{array}{cc}2& 1\\ -1& 0\\ 3& -2\end{array}\right]$

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

$A{A}^T=\left[\begin{array}{ccc}2& -1& 3\\ 1& 0& -2\end{array}\right]\left[\begin{array}{cc}2& 1\\ -1& 0\\ 3& -2\end{array}\right]$

$=\left[\begin{array}{cc}(2\times )2+(-1)\times (-1)+3\times 3& (2\times 1+(-1)\times 0+3\times (-2))\\ (1\times 2+0\times (-1)+(-2)\times 3)& (1\times 1+0\times 0+(-2)\times (-2))\end{array}\right]$ 

$=\left[\begin{array}{cc}4+1+9& 2+0-6\\ 2+0-6& 1+0+4\end{array}\right]=\left[\begin{array}{cc}14& -4\\ -4& 5\end{array}\right]$

Example 6: Let $A=\left[\begin{array}{cc}1& 2\\ 0& 1\end{array}\right],B=\left[\begin{array}{cc}x& y\\ z& w\end{array}\right]$. Find the condition(s) on x, y, z, w such that AB = BA.

Solution:

Compute both:

$AB=\left[\begin{array}{cc}1\times x+2\times z& 1\times y+2\times w\\ 0\times x+1\times z& 0\times y+1\times w\end{array}\right]=\left[\begin{array}{cc}x+2z& y+2w\\ z& w\end{array}\right]$ 

$BA=\left[\begin{array}{cc}x\times 1+y\times 0& x\times 2+y\times 1\\ z\times 1+w\times 0& z\times 2+w\times 1\end{array}\right]=\left[\begin{array}{cc}x& 2x+y\\ z& 2z+w\end{array}\right]$

Equating AB = BA:

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

From equations:

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

$y+2w=2x+y$

$\Rightarrow 2w=2x$

$\Rightarrow w=x$

Now,

$w=2z+w$

$\Rightarrow 0=2z$

$\Rightarrow z=0\text{(already satisfied)}$

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

3.0Practice Matrix Multiplication Questions

Question 1: If $A=\left[\begin{array}{cc}1& 0\\ -1& 2\end{array}\right],B=\left[\begin{array}{cc}3& 4\\ 0& 1\end{array}\right]$. Find AB and BA.

Question 2: Find the product of  $A=\left[\begin{array}{cc}2& -1\\ 0& 3\end{array}\right],B=\left[\begin{array}{cc}4& 0\\ 5& 1\end{array}\right]$

Question 3: If $A=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right],B=\left[\begin{array}{cc}1& 0\\ 0& 0\end{array}\right]$. Then what is AB?

Question 4: Verify whether matrix multiplication is associative for the following matrices:$A=\left[\begin{array}{cc}1& 2\end{array}\right],B=\left[\begin{array}{c}0\\ 1\end{array}\right],C=[3]$. Check if A(BC) = (AB)C

Also Practice : Matrices and Determinants previous year questions with solutions

4.0Tips for Solving Matrix Multiplication Questions

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

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