2x2 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 2x2 matrices, provide examples of how to perform these operations, and solve some practice problems.
How to add and subtract 2x2 matrices?
To add or subtract 2x2 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 2x2 matrices:
$$A = \begin{bmatrix} 1& 2\\ 3& 4\end{bmatrix}$$
$$B = \begin{bmatrix} 5& 6\\ 7& 8\end{bmatrix}$$
To add these matrices, simply add the corresponding entries:
$$A+B = \begin{bmatrix} 1+5& 2+6\\ 3+7& 4+8\end{bmatrix}$$
$$A+B = \begin{bmatrix} 6& 8\\ 10& 12\end{bmatrix}$$
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 2x2 arrays.
Solved exercises on addition and subtraction of 2x2 matrices
EXAMPLE 1
Find the addition of the following matrices:
$$A = \begin{bmatrix} 2& 3\\ 4& 5\end{bmatrix}$$
$$B = \begin{bmatrix} 1& 1\\ 2& 2\end{bmatrix}$$
Solution
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:
$$A = \begin{bmatrix} 2& 3\\ 4& 5\end{bmatrix}$$
$$B = \begin{bmatrix} 1& 1\\ 2& 2\end{bmatrix}$$
Solution
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?
$$A = \begin{bmatrix} 5& 7\\ 1& 8\end{bmatrix}$$
$$B = \begin{bmatrix} 6& 2\\ 9& 3\end{bmatrix}$$
Solution
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:
$$A = \begin{bmatrix} 2 & 0 \\ 0 & 3 \end{bmatrix} $$
$$B = \begin{bmatrix} 4 & 0 \\ 0 & 4 \end{bmatrix}$$
Solution
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:
$$A = \begin{bmatrix} -3 & 4 \\ 2 & -1 \end{bmatrix}$$
$$B = \begin{bmatrix} 1 & -4 \\ -2 & 3 \end{bmatrix}$$
Solution
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:
$$A = \begin{bmatrix} -8 & a \\ 6 & -5 \end{bmatrix}$$
$$B = \begin{bmatrix} 2 & -5 \\ b & 3 \end{bmatrix}$$
$$A+B = \begin{bmatrix} -6 & -1 \\ -2 & -2 \end{bmatrix}$$
Solution
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 $latex n$?
$$A = \begin{bmatrix} 7 & 12 \\ 7 & n\end{bmatrix}$$
$$B = \begin{bmatrix} -m & 3 \\ -7 & -13 \end{bmatrix}$$
$$B-A = \begin{bmatrix} -4 & -9 \\ -14 & -16 \end{bmatrix}$$
Solution
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:
$$A = \begin{bmatrix} -2 & 5 \\ 1 & -4 \end{bmatrix}$$
$$B = \begin{bmatrix} 3 & -6 \\ -4 & 7 \end{bmatrix}$$
$$C = \begin{bmatrix} -1 & 0 \\ 2 & 3 \end{bmatrix}$$
Solution
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 2x2 matrices - Practice problems
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See also
Interested in learning more about matrices? You can take a look at these pages:
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