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 = [Tex]\begin{pmatrix} 18 \\ 15 \\ -21 \end{pmatrix} [/Tex]

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)

= [Tex]\begin{pmatrix} -6 & -5 & 7 \end{pmatrix} [/Tex]

Problem 2: Find the product of A and B.

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

Solution:

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

Product Matrix [Tex]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} [/Tex]

[Tex]AB = \begin{pmatrix} -1 & 6 \\ 2 & 8 \end{pmatrix} [/Tex]

Problem 3: Find the product of the following matrices:

[Tex]A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 2 & 1 \\ 1 & 2 & 5 \end{pmatrix} [/Tex]

[Tex]B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 2 & 1 \end{pmatrix} [/Tex]

Solution:

Given

[Tex]A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 2 & 1 \\ 1 & 2 & 5 \end{pmatrix} [/Tex]

[Tex]B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 2 & 1 \end{pmatrix} [/Tex]

Then,

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

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

= [Tex]\begin{pmatrix} 7 & 5 \\ 2 & 3 \\ 11 & 7 \end{pmatrix} [/Tex]

Matrix Multiplication: Worksheet

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

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

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

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

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

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

FAQs on Matrix Multiplication

What is a Matrix?

A matrix is a rectangular array of numbers arranged in rows and columns. The numbers in the matrix are called elements or entries.

Is Matrix Multiplication Distributive?

Yes, matrix multiplication is distributive over matrix addition. This means

A⋅(B+C)=A⋅B+A⋅C and (A+B)⋅C=A⋅C+B⋅C.

What is Matrix Multiplication?

Matrix multiplication involves multiplying two matrices to produce a third matrix. For matrices A (of dimensions m×n) and B (of dimensions n×p), their product C=A⋅B results in a matrix C of dimensions m×p.

What Are Some Common Applications?

• Transformations in Graphics: Rotation, scaling, and translation of objects.
• Systems of Linear Equations: Represented in matrix form A⋅x=b.
• Data Analysis: Principal Component Analysis (PCA) and other statistical computations.
• Machine Learning: Operations in neural networks and optimization.