How to Multiply 2 × 2 Matrices

Last Updated : 23 Jul, 2025

Matrix multiplication is a mathematical operation that combines two matrices to produce a new matrix, also known as the product matrix. This operation is defined for two matrices, A and B, if the number of columns in matrix A is equal to the number of rows in matrix B.

To multiply any two matrices, we can use the following steps:

  • Pick a row from the first matrix (Matrix A) and a column from the second matrix (Matrix B).
  • Multiply each element in the row by the corresponding element in the column.
  • Add up all those products to get a single number.
  • That number becomes the entry in the new matrix at the position corresponding to that row and column.

Repeat this for all rows and column to get your answer.


Note: Two matrices A and B can be multiplied if the number of columns in A equals the number of rows in B. If A is m × n and B is n × o, then the product AB is m × o.

  • Multiplying a 2 × 3 matrix by a 3 × 2 matrix is valid, resulting in a 2 × 2 matrix.
  • A 3 × 3 matrix cannot be multiplied by a 4 × 2 matrix because the dimensions do not align for multiplication.

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Solved Example on Multiplying 2 × 2 Matrices

Example 1: Let A = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} and B =\begin{bmatrix} 5 & 7 \\ 6 & 8 \end{bmatrix}, then find A × B.

Solution:

A \times B = \begin{bmatrix} 1 & 3 \\ 2 & 4 \end{bmatrix} \times \begin{bmatrix} 5 & 7 \\ 6 & 8 \end{bmatrix}

= \begin{bmatrix} 1 \times 5 + 3 \times 6 & 1 \times 7 + 3 \times 8 \\ 2 \times 5 + 4 \times 6 & 2 \times 7 + 4 \times 8 \end{bmatrix}

= \begin{bmatrix} 5 + 18 & 7 + 24 \\ 10 + 24 & 14 + 32 \end{bmatrix}

= \begin{bmatrix} 23 & 31 \\ 34 & 46 \end{bmatrix}

Example 2: Let A = \begin{bmatrix} -1 & 2 \\ 3 & 0 \end{bmatrix} and B =\begin{bmatrix} 4 & 5 \\ -3 & 6 \end{bmatrix} , then find A × B.

Solution:

A × B =\begin{bmatrix} -1 & 2 \\ 3 & 0 \end{bmatrix} \times \begin{bmatrix} 4 & 5 \\ -3 & 6 \end{bmatrix}

= \begin{bmatrix} (-1 \times 4 + 2 \times -3) & (-1 \times 5 + 2 \times 6) \\ (3 \times 4 + 0 \times -3) & (3 \times 5 + 0 \times 6) \end{bmatrix}

= \begin{bmatrix} (-4 -6 ) & (-5 + 12) \\ (12 + 0) & (15 + 0) \end{bmatrix}

= \begin{bmatrix} 6 & 15 \\ 12 & -9 \end{bmatrix}

Example 3: Let B = \begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix} and B2 = qB. Find the value of q.

Solution:

Calculating B2

B2 = B × B = \begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix} \times \begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix}

= \begin{bmatrix} (3 \times 3 + 5 \times 1) & (3 \times 5 + 5 \times 2) \\ (1 \times 3 + 2 \times 1) & (1 \times 5 + 2 \times 2) \end{bmatrix}

= \begin{bmatrix} 9 + 5 & 15 + 10 \\ 3 + 2 & 5 + 4 \end{bmatrix}

= \begin{bmatrix} 14 & 25 \\ 5 & 9 \end{bmatrix}

comparing the result with qB .

we get, \text q =1.

Practice Questions on Multiplying 2 × 2 Matrices

Question 1: Let's A = \begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 1 & 3 \\ 5 & 2 \end{bmatrix}.Find A × B?

Question 2: Let's A = \begin{bmatrix} 4 & -2 \\ 0 & 1 \end{bmatrix}, \quad B = \begin{bmatrix} 2 & 5 \\ 3 & 1 \end{bmatrix}. Find A × B?

Question 3: Let's A = \begin{bmatrix} 3 & 4 \\ 2 & 1 \end{bmatrix}. Find A2?

Question 4: Let's A = \begin{bmatrix} 2 & 3 \\ -1 & 5 \end{bmatrix} \, \text{and} \, k = 3. Find kA?

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