# Addition and Subtraction of 2×2 Matrices with Examples

2×2 matrices are matrices with two rows and two columns. Adding and subtracting this type of matrices is very simple, since we only have to add or subtract the corresponding entries of the matrices.

In this article, we will explore the rules for adding and subtracting 2×2 matrices, provide examples of how to perform these operations, and solve some practice problems.

##### LINEAR ALGEBRA

Relevant for

Learning to add and subtract matrices with examples.

See examples

##### LINEAR ALGEBRA

Relevant for

Learning to add and subtract matrices with examples.

See examples

## How to add and subtract 2×2 matrices?

To add or subtract 2×2 matrices, the corresponding entries of each matrix must be added or subtracted.

This means that we only have to add or subtract the elements that are in the same positions in the matrices.

For example, consider two 2×2 matrices:

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

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

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

$$A+B = \begin{bmatrix} 6& 8\\ 10& 12\end{bmatrix}$$

To subtract the matrices, we subtract the corresponding entries:

$$A-B = \begin{bmatrix} 1-5& 2-6\\ 3-7& 4-8\end{bmatrix}$$

$$A-B = \begin{bmatrix} -4& -4\\ -4& -4\end{bmatrix}$$

Note: Note that you can only add or subtract arrays of the same size, so in this case, the two arrays must be 2×2 arrays.

## Solved exercises on addition and subtraction of 2×2 matrices

### EXAMPLE 1

Find the addition of the following matrices:

$$A = \begin{bmatrix} 2& 3\\ 4& 5\end{bmatrix}$$

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

To solve this addition, we only have to add the corresponding entries:

$$A+B = \begin{bmatrix} 2+1& 3+1\\ 4+2& 5+2\end{bmatrix}$$

$$A+B = \begin{bmatrix} 3& 4\\ 6& 7\end{bmatrix}$$

### EXAMPLE 2

Determine the subtraction A-B of the following matrices:

$$A = \begin{bmatrix} 2& 3\\ 4& 5\end{bmatrix}$$

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

In this case, we subtract the corresponding elements of the B matrix from the corresponding elements of the A matrix:

$$A-B = \begin{bmatrix} 2-1& 3-1\\ 4-2& 5-2\end{bmatrix}$$

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

### EXAMPLE 3

What is the result of subtraction B-A?

$$A = \begin{bmatrix} 5& 7\\ 1& 8\end{bmatrix}$$

$$B = \begin{bmatrix} 6& 2\\ 9& 3\end{bmatrix}$$

Note that subtraction B-A is not equal to subtraction A-B. We must place the matrices in the correct order to obtain the result:

$$B-A = \begin{bmatrix} 6-5& 2-7\\ 9-1& 3-8\end{bmatrix}$$

$$B-A = \begin{bmatrix} 1& -5\\ 8& -5\end{bmatrix}$$

### EXAMPLE 4

Find the result of adding and subtracting (A-B) the following matrices:

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

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

Adding the two given matrices, we have:

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

$$A+B = \begin{bmatrix} 6 & 0 \\ 0 & 7 \end{bmatrix}$$

Subtracting the two matrices, we have:

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

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

### EXAMPLE 5

Find the addition and difference A-B of the following matrices:

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

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

When we add matrices A and B, we have:

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

$$A+B= \begin{bmatrix} -2 & 0 \\ 0 & 2 \end{bmatrix}$$

When we subtract matrix B from matrix A, we have:

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

$$A-B= \begin{bmatrix} -4 & 8 \\ 4 & -4 \end{bmatrix}$$

### EXAMPLE 6

Find the values of $latex a$ and $latex b$ given the following information:

$$A = \begin{bmatrix} -8 & a \\ 6 & -5 \end{bmatrix}$$

$$B = \begin{bmatrix} 2 & -5 \\ b & 3 \end{bmatrix}$$

$$A+B = \begin{bmatrix} -6 & -1 \\ -2 & -2 \end{bmatrix}$$

This exercise is different from the previous ones. In this case, we have to determine the value of an element in each matrix knowing the sum of the matrices.

To solve this, we form the sum at each required position:

$latex a+(-5)=-1$

$latex 6+b=-2$

Now we solve the equations:

$latex a=4$

$latex b=-8$

### EXAMPLE 7

What are the values of $latex m$ and $latex n$?

$$A = \begin{bmatrix} 7 & 12 \\ 7 & n\end{bmatrix}$$

$$B = \begin{bmatrix} -m & 3 \\ -7 & -13 \end{bmatrix}$$

$$B-A = \begin{bmatrix} -4 & -9 \\ -14 & -16 \end{bmatrix}$$

Similar to the previous exercise, we will form an equation with each required position:

$latex -4-(-m)=7$

$latex -16-(-13)=n$

Now we solve the equations:

$latex m=11$

$latex n=-3$

### EXAMPLE 8

Find the result of A+B-C:

$$A = \begin{bmatrix} -2 & 5 \\ 1 & -4 \end{bmatrix}$$

$$B = \begin{bmatrix} 3 & -6 \\ -4 & 7 \end{bmatrix}$$

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

By the associative property of the addition of matrices, we can write (A + B – C) = (A + B) – C.

Then, we start by finding the result of the sum A+B:

$$(A + B) = \begin{bmatrix} -2+3 & 5+(-6) \ 1+(-4) & -4+7 \end{bmatrix}$$

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

Then, we can subtract C from (A+B):

$$(A + B) – C = \begin{bmatrix} 1-(-1) & -1-0 \ -3-2 & 3-3 \end{bmatrix}$$

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

## Addition and subtraction of 2×2 matrices – Practice problems

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