Multiplication of 2x2 Matrices - Examples with Answers

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:

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

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

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.

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:

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

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:

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

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:

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

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:

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

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:

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

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:

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

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 $latex m_{22}$:

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

$latex 3=b$

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 $latex m_{21}$:

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

$latex 3=-5 + 4q$

$latex 4q=8$

$latex q=2$