# Multiplying the 2×2 Matrices

## Definition

A mathematical operation of multiplying a two by two matrix by another two by two matrix is called the multiplication of two by two matrices.

### Introduction

The two by two matrices are involved in multiplication. So, it is essential for everyone to learn how to multiply $2 \times 2$ matrices in mathematics. There is a special procedure for multiplying the matrices of order $2$ and it is called the multiplication of $2 \times 2$ matrix.

Now, let’s learn how to multiply the $2 \times 2$ matrices mathematically from an example with understandable step by step process.

${\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{bmatrix}}$ $\times$ ${\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ \end{bmatrix}}$

#### Procedure

There are two simple steps to multiply a $2 \times 2$ square matrix by another square matrix of order $2$.

1. Multiply the two entries in each row of a matrix by the two entries of each column of another matrix respectively.
2. Add a product of two entries to the product of another elements, and then write the sum of them as an element in the respective position of the matrix.

#### Example

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

#### Formula

The $2 \times 2$ matrix multiplication formula is expressed in the following algebraic equation form.

${\begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ \end{bmatrix}}$ $\times$ ${\begin{bmatrix} b_{11} & b_{12} \\ b_{21} & b_{22} \\ \end{bmatrix}}$ $\,=\,$ ${\begin{bmatrix} a_{11}b_{11}+a_{12}b_{21} & a_{11}b_{12}+a_{12}b_{22} \\ a_{21}b_{11}+a_{22}b_{21} & a_{21}b_{12}+a_{22}b_{22} \\ \end{bmatrix}}$

Learn the multiplication of two by two matrix formula with proof to learn how to derive the $2 \times 2$ matrix multiplication formula in mathematics.

##### Worksheet

$(1).\,\,$ Evaluate ${\begin{bmatrix} -2 & 3 \\ -1 & 4 \\ \end{bmatrix}}$ $\times$ ${\begin{bmatrix} 6 & 4 \\ 3 & -1 \\ \end{bmatrix}}$

$(2).\,\,$ Evaluate ${\begin{bmatrix} 2 & -5 \\ 0 & -3 \\ \end{bmatrix}}$ $\times$ ${\begin{bmatrix} 1 & -1 \\ 3 & 2 \\ \end{bmatrix}}$

$(3).\,\,$ Evaluate ${\begin{bmatrix} 1 & 3 \\ 3 & 2 \\ \end{bmatrix}}$ $\times$ ${\begin{bmatrix} -2 & 3 \\ -4 & 1 \\ \end{bmatrix}}$

The questions on $2 \times 2$ matrix multiplication for practice and example problems with solutions to learn how to multiply two or more $2$ by $2$ matrix in mathematics.

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Jun 26, 2023

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