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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