Matrix Multiplication Practice Questions

A matrix is a set of numbers arranged in rows and columns to form a rectangular array. Multiplying a matrix by another matrix is called "matrix multiplication". In this article, we will learn what matrix multiplication is. And practice some questions related to it.

What is Matrix Multiplication?

In linear algebra, a matrix is an arrangement of elements in the form of rows and columns. An m × n matrix has m rows and n columns. If a matrix has an equal number of rows and columns, then the matrix is called a square matrix. When we perform multiplication operations with matrices, it is called matrix multiplication.

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Matrix Multiplication Practice Questions with Solution

Problem 1: If the matrix A = \begin{pmatrix} 18 \\ 15 \\ -21 \end{pmatrix}

then what is the scalar multiple (-1/3)A?

Solution:

To find (-1/3) A, we have to multiply every element of A by (-1/3). Then

(-1/3) A= 18 x (-1/3) 15 x (-1/3) -21 x (-1/3)

= \begin{pmatrix} -6 & -5 & 7 \end{pmatrix}

Problem 2: Find the product of A and B.

A = \begin{pmatrix} 3 & 2 & -1 \\ 4 & 2 & 0 \end{pmatrix} and B = \begin{pmatrix} 0 & 1 \\ 1 & 2 \\ 3 & 1 \end{pmatrix}

Solution:

Given A = \begin{pmatrix} 3 & 2 & -1 \\ 4 & 2 & 0 \end{pmatrix} and B = \begin{pmatrix} 0 & 1 \\ 1 & 2 \\ 3 & 1 \end{pmatrix}

Product Matrix AB = \begin{pmatrix} 3 \cdot 0 + 2 \cdot 1 + (-1) \cdot 3 & 3 \cdot 1 + 2 \cdot 2 + (-1) \cdot 1 \\ 4 \cdot 0 + 2 \cdot 1 + 0 \cdot 3 & 4 \cdot 1 + 2 \cdot 2 + 0 \cdot 1 \end{pmatrix}

AB = \begin{pmatrix} -1 & 6 \\ 2 & 8 \end{pmatrix}

Problem 3: Find the product of the following matrices:

A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 2 & 1 \\ 1 & 2 & 5 \end{pmatrix}

B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 2 & 1 \end{pmatrix}

Solution:

Given

A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 2 & 1 \\ 1 & 2 & 5 \end{pmatrix}

B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 2 & 1 \end{pmatrix}

Then,

A * B = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 2 & 1 \\ 1 & 2 & 5 \end{pmatrix} * \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 2 & 1 \end{pmatrix}

= \begin{pmatrix} 1 + 0 + 6 & 0 + 2 + 3 \\ 0 + 0 + 2 & 0 + 2 + 1 \\ 1 + 0 + 10 & 0 + 2 + 5 \end{pmatrix}

= \begin{pmatrix} 7 & 5 \\ 2 & 3 \\ 11 & 7 \end{pmatrix}

Matrix Multiplication: Worksheet

1. Given matrices: A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} and B = \begin{pmatrix} 2 & 0 \\ 1 & 3 \end{pmatrix}. Find AB

2. Given matrices: C = \begin{pmatrix} 5 & -1 \\ 2 & 3 \end{pmatrix} and D = \begin{pmatrix} 0 & 4 \\ -2 & 1 \end{pmatrix}. Find CD.

3. Given matrices: E = \begin{pmatrix} 3 & 0 & 2 \\ 1 & 4 & 5 \end{pmatrix} and F = \begin{pmatrix} 2 & 3 \\ 0 & 1 \\ 1 & 4 \end{pmatrix}. Find EF.

4. Given matrices: G = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix} and H = \begin{pmatrix} 7 & 8 \\ 9 & 10 \\ 11 & 12 \end{pmatrix}.​​ Find GH.

5. Given matrices: I = \begin{pmatrix} 2 & 4 \\ 6 & 8 \end{pmatrix} and J = \begin{pmatrix} 1 & 3 \\ 5 & 7 \end{pmatrix}. Find IJ.

6. Given matrices: M = \begin{pmatrix} 1 & 0 & 2 \\ -1 & 3 & 1 \end{pmatrix} and N = \begin{pmatrix} 4 & 1 \\ 2 & 2 \\ 0 & 3 \end{pmatrix}. Find MN.