# Multiplication of 2×2 Matrices – Examples with Answers

Multiplication of 2×2 matrices is a fundamental operation of linear algebra that has numerous applications. To find each element of the resulting matrix, we multiply each row of the first matrix by the corresponding columns of the second matrix and add the products.

In this article, we will explore the key concepts and techniques for solving 2×2 matrix multiplication. We will look at several exercises to master the concepts.

##### LINEAR ALGEBRA

Relevant for

Learning about 2×2 matrix multiplication with examples.

See examples

##### LINEAR ALGEBRA

Relevant for

Learning about 2×2 matrix multiplication with examples.

See examples

## How to multiply 2×2 matrices?

To multiply two 2×2 matrices, we have to multiply the rows of the first matrix by the corresponding columns of the second matrix. The products are then added to find each element.

We can multiply two 2×2 matrices by following the steps below:

Step 1: Write the matrices in the following way:

$$A=\begin{bmatrix} a & b \\ c & d \end{bmatrix}$$

$$B=\begin{bmatrix} e & f \\ g & h \end{bmatrix}$$

Step 2: To find the first element of the resulting matrix, multiply the first row of the first matrix by the first column of the second matrix and add the products. For example:

$latex a \times e + b \times g = x$

The resulting matrix will have dimensions 2×2, so it will have four elements.

Step 3: To find the remaining elements, repeat the process with the remaining rows and columns, and fill the elements of the resulting matrix as follows:

$$A\times B= \begin{bmatrix} a \times e + b \times g & a \times f + b \times h \\ c \times e + d \times g & c \times f + d \times h \\ \end{bmatrix}$$

## Solved exercises on multiplication of 2×2 matrices

### EXAMPLE 1

Find the product A×B:

$$A=\begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$$

$$B=\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$$

Suppose our product is the matrix M = A×B.

We can find the term $latex m_{11}$ by multiplying the first row of the matrix A by the first column of the matrix B. Then, we have:

$latex m_{11}=2 \times 5 + 1 \times 7$

For the term $latex m_{12}$, we multiply the first row of matrix A by the second column of matrix B:

$latex m_{12}=2 \times 6 + 1 \times 8$

The term $latex m_{21}$ is found by multiplying the second row of matrix A by the first column of matrix B:

$latex m_{21}=3 \times 5 + 4 \times 7$

And the term $latex m_{22}$ is found by multiplying the second row of matrix A by the second column of matrix B:

$latex m_{22}=3 \times 6 + 4 \times 8$

So, we have:

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

$$M=\begin{bmatrix} 17 & 22 \\ 38 & 53 \end{bmatrix}$$

### EXAMPLE 2

What is the product of the multiplication B×A?

$$A=\begin{bmatrix} 2 & 1 \\ 3 & 4 \end{bmatrix}$$

$$B=\begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$$

Now that we know how to calculate each term of the matrix resulting from the multiplication, we can simply write like this:

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

$$M=\begin{bmatrix} 28 & 29 \\ 46 & 51 \end{bmatrix}$$

We can observe by comparing examples 1 and 2, that the multiplication A×B is not equal to the multiplication B×A. That is, the order does matter.

### EXAMPLE 3

Find the multiplication A×B with the following matrices:

$$A=\begin{bmatrix} -1 & 3 \\ 4 & 2 \end{bmatrix}$$

$$B=\begin{bmatrix} 0 & 5 \\ -2 & 1 \end{bmatrix}$$

Find each element of the resulting matrix:

$$M=\begin{bmatrix} -1 \times 0 + 3 \times -2 & -1 \times 5 + 3 \times 1 \\ 4 \times 0 + 2 \times -2 & 4 \times 5 + 2 \times 1 \end{bmatrix}$$

Simplifying this, we have:

$$M=\begin{bmatrix} -6 & -2 \\ -4 & 22 \end{bmatrix}$$

### EXAMPLE 4

Multiply the following matrices to find the product A×B

$$A=\begin{bmatrix} 2 & -3 \\ 5 & 1 \end{bmatrix}$$

$$B=\begin{bmatrix} 0 & 4 \\ 2 & 1 \end{bmatrix}$$

Multiply the rows of the first matrix by the corresponding columns of the second matrix:

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

Now, we simplify the operations and we have:

$$M=\begin{bmatrix} -6 & 5 \\ 2 & 21 \end{bmatrix}$$

### EXAMPLE 5

Find the product A×B considering the following matrices:

$$A=\begin{bmatrix} 2 & 1 \\ 3 & -1 \end{bmatrix}$$

$$B=\begin{bmatrix} 0 & 4 \\ -2 & 3 \end{bmatrix}$$

To find each element of the resulting matrix, we write as follows:

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

Now, we just simplify:

$$M=\begin{bmatrix} -2 & 11 \\ 2 & 9 \end{bmatrix}$$

### EXAMPLE 6

Multiply the matrices to find A×B:

$$A=\begin{bmatrix} 4 & -1 \\ 2 & 3 \end{bmatrix}$$

$$B=\begin{bmatrix} 0 & 2 \\ 1 & 5 \end{bmatrix}$$

Multiply the rows of the first matrix by the corresponding columns of the second matrix:

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

By simplifying, we are left with:

$$M=\begin{bmatrix} -1 & 3 \\ 3 & 16 \end{bmatrix}$$

### EXAMPLE 7

Solve the multiplication A×B with the following matrices:

$$A=\begin{bmatrix} 1 & -2 \\ 3 & 0 \end{bmatrix}$$

$$B=\begin{bmatrix} 0 & 1 \\ 2 & 4 \end{bmatrix}$$

Find each term of the new matrix as follows:

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

And the matrix product is:

$$M=\begin{bmatrix} -4 & -7 \\ 0 & 3 \end{bmatrix}$$

### EXAMPLE 8

What is the product A×B?

$$A=\begin{bmatrix} 3 & -1 \\ -2 & 5 \end{bmatrix}$$

$$B=\begin{bmatrix} 1 & 2 \\ 0 & -3 \end{bmatrix}$$

To find each element of the resulting matrix, we have:

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

And simplifying:

$$M=\begin{bmatrix} 3 & 9 \\ -2 & -13 \end{bmatrix}$$

### EXAMPLE 9

If we have the following, what are the values of $latex a$ and $latex b$?

$$A=\begin{bmatrix} 2 & 1 \\ 1 & b \end{bmatrix}$$

$$B=\begin{bmatrix} a & 0 \\ 2 & 1 \end{bmatrix}$$

$$A\times B= \begin{bmatrix} 10 & 1 \\ 10 & 3 \end{bmatrix}$$

To find the value of $latex a$, we can form an equation with the necessary process to obtain the $latex m_{11}$ element of the resulting matrix.

That is, we have:

$latex m_{11}=2 \times a + 1 \times 2$

$latex 10=2 \times a + 1 \times 2$

Solving for $latex a$:

$latex 10=2a+2$

$latex 2a=8$

$latex a=4$

Now, we form an equation with the process required to find $latex m_{22}$:

$latex m_{22}=1 \times 0 + b \times 1$

$latex 3=b$

### EXAMPLE 10

Find the values of $latex p$ and $latex q$ considering the following:

$$A=\begin{bmatrix} 2 & p\\ -1 & 4 \end{bmatrix}$$

$$B=\begin{bmatrix} 5 & 1\\ q & -3 \end{bmatrix}$$

$$A\times B= \begin{bmatrix} 16 & -7 \\ 3 & -13 \end{bmatrix}$$

Create an equation for $latex m_{12}$:

$latex m_{12}=2 \times 1 + p \times -3$

$latex -7=2 + -3p$

Solving for $latex p$:

$latex -3p=-9$

$latex p=3$

Now, we form an equation for $latex m_{21}$:

$latex m_{21}=-1 \times 5 + 4 \times q$

$latex 3=-5 + 4q$

$latex 4q=8$

$latex q=2$

## 2×2 Matrix Multiplication – Practice problems

You have completed the quiz!

Write the answer in the input box.

$latex =$
const quiz2 = { question: “¿Cuál es el término 18 de la sucesión aritmética en la que el término 4 es 17 y el término 8 es $latex x^2-13$?”, symbol: “$latex a_{18}=$”, answers: [“-67”, “- 67”], answerUnit: “km”, }

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