How to Add and Subtract Matrices

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Matrices operations

The basic operations on a matrix are addition, subtraction, and multiplication. For matrices to be added or subtracted, they must be in the same order. For matrices to be multiplied, the number of columns in the first Matrix must equal the number of rows in the second Matrix.

Adding Matrices

If \(A[a_{ij}]_{m \times n}\) and \(B[b_{ij}]_{m \times n}\) are two matrices with the same order, then \(A \ + \ B\) is a matrix, and each matrix element is the sum of the corresponding parts. i.e., \(A \ + \ B \ = \ [a_{ij} \ + \ b_{ij}]_{m \times n}\)
Consider \(A\) and \(B\) with a \(2 \times 2\) order. Then the total can be found by:

\(\begin{bmatrix}a_1 & a_2\\\ a_3 & a_4\end{bmatrix} + \begin{bmatrix}b_1 & b_2\\\ b_3 & b_4\end{bmatrix} = \begin{bmatrix}a_1 \ + \ b_1 & a_2 \ + \ b_2\\\ a_3 \ + \ b_3 & a_4 \ + \ b_4\end{bmatrix}\)

Properties for Matrix Addition

If \(A\), \(B\), and \(C\) are all in the same order, then,

The Commutative Law says: \(A \ + \ B \ = \ B \ + \ A\)
The Associative Law:: \((A \ + \ B) \ + \ C \ = \ A \ + \ (B \ + \ C)\)
Identity of the Matrix: \(A \ + \ 0 \ = \ 0 \ + \ A \ = \ 0\) (\(0\) is zero matrix which is additive identity of the matrix)
Additive Inverse: \(A \ + \ (-A) \ = \ (-A) \ + \ A \ = \ 0\), where \((-A)\) is the result of changing the sign of each element of \(A\) which is the additive inverse of the Matrix.
\(\begin{equation} {A \ + \ B \ = \ A \ + \ C \\\ B \ + \ A \ = \ C \ + \ A} \end{equation} \ ⇒ \ B \ = \ C\)
\(tr(A \ ± \ B \ = \ tr(A) \ ± \ tr(B)\)
If \(A \ + \ B \ = \ 0 \ = \ B \ + \ A\), then \(B\) is the additive inverse of \(A\), and \(A\) is the additive inverse of \(B\).

Subtracting Matrices

We can say that if \(A\) and \(B\) are both matrices with the same order: \(A \ - \ B \ = \ A \ + \ (-B)\)

Consider the two \(2 \times 2\) matrices, \(A\) and \(B\). The difference can then be found by:

\(\begin{bmatrix}a_1 & a_2\\\ a_3 & a_4\end{bmatrix} - \begin{bmatrix}b_1 & b_2\\\ b_3 & b_4\end{bmatrix} = \begin{bmatrix}a_1 \ - \ b_1 & a_2 \ - \ b_2\\\ a_3 \ - \ b_3 & a_4 \ - \ b_4\end{bmatrix}\)

We can subtract the matrices by subtracting each element of one Matrix from the corresponding element of the other Matrix. I.e., \(A \ - \ B \ = \ [a_{ij} \ - \ b_{ij}]_{m \times n}\)

Free printable Worksheets

Exercises for Adding and Subtracting Matrices

1) Find the answer: \(\begin{bmatrix} 2 & -5 \\\ 7 & 3 \end{bmatrix} + \begin{bmatrix} -4 & -2 \\\ 4 & -9 \end{bmatrix}\)

2) Find the answer: \(\begin{bmatrix} -3 & 2 \\\ 12 & -6 \end{bmatrix} + \begin{bmatrix} 5 & 13 \\\ -7 & 8 \end{bmatrix}\)

3) Find the answer: \(\begin{bmatrix} 15 & -21 \\\ -8 & 13 \end{bmatrix} + \begin{bmatrix} -7 & 14 \\\ -9 & 4 \end{bmatrix}\)

4) Find the answer: \(\begin{bmatrix} 24 & 31 \\\ -29 & 25 \end{bmatrix} + \begin{bmatrix} -17 & -26 \\\ 14 & 9 \end{bmatrix}\)

5) Find the answer: \(\begin{bmatrix} 9 & 11 \\\ -14 & 5 \\\ 15 & 21 \end{bmatrix} + \begin{bmatrix} -18 & -7 \\\ 14 & -18 \\\ -29 & 3 \end{bmatrix}\)

6) Find the answer: \(\begin{bmatrix} 2 & -5 \\\ 7 & 3 \end{bmatrix} - \begin{bmatrix} -4 & -2 \\\ 4 & -9 \end{bmatrix}\)

7) Find the answer: \(\begin{bmatrix} 17 & -28 \\\ -15 & 32 \end{bmatrix} - \begin{bmatrix} 12 & -5 \\\ 9 & -11 \end{bmatrix}\)

8) Find the answer: \(\begin{bmatrix} 9 & -7 \\\ 18 & -19 \end{bmatrix} - \begin{bmatrix} -3 & 9 \\\ 6 & 4 \end{bmatrix}\)

9) Find the answer: \(\begin{bmatrix} 11 & 19 \\\ 27 & 34 \end{bmatrix} - \begin{bmatrix} -42 & -7 \\\ 5 & 39 \end{bmatrix}\)

10) Find the answer: \(\begin{bmatrix} 18 & -14 \\\ 53 & -34 \\\ -13 & -15 \end{bmatrix} - \begin{bmatrix} 91 & -47 \\\ 19 & -20 \\\ 30 & -16 \end{bmatrix}\)

Answers

1) Find the answer: \(\begin{bmatrix} 2 & -5 \\\ 7 & 3 \end{bmatrix} + \begin{bmatrix} -4 & -2 \\\ 4 & -9 \end{bmatrix}\)

\(\color{red}{\begin{bmatrix} 2 & -5 \\\ 7 & 3 \end{bmatrix} + \begin{bmatrix} -4 & -2 \\\ 4 & -9 \end{bmatrix} \ = \ \begin{bmatrix} 2 \ + \ (-4) & -5 \ + \ (-2) \\\ 7 \ + \ 4 & 3 \ + \ (-9) \end{bmatrix} \ = \ \begin{bmatrix} -2 & -7 \\\ 11 & -6 \end{bmatrix}}\)

2) Find the answer: \(\begin{bmatrix} -3 & 2 \\\ 12 & -6 \end{bmatrix} + \begin{bmatrix} 5 & 13 \\\ -7 & 8 \end{bmatrix}\)

\(\color{red}{\begin{bmatrix} -3 & 2 \\\ 12 & -6 \end{bmatrix} + \begin{bmatrix} 5 & 13 \\\ -7 & 8 \end{bmatrix} \ = \ \begin{bmatrix} -3 \ + \ 5 & 2 \ + \ 13 \\\ 12 \ + \ (-7) & -6 \ + \ 8 \end{bmatrix} \ = \ \begin{bmatrix} 2 & 15 \\\ 5 & 2 \end{bmatrix}}\)

3) Find the answer: \(\begin{bmatrix} 15 & -21 \\\ -8 & 13 \end{bmatrix} + \begin{bmatrix} -7 & 14 \\\ -9 & 4 \end{bmatrix}\)

\(\color{red}{\begin{bmatrix} 15 & -21 \\\ -8 & 13 \end{bmatrix} + \begin{bmatrix} -7 & 14 \\\ -9 & 4 \end{bmatrix} \ = \ \begin{bmatrix} 15 \ + \ (-7) & -21 \ + \ 14 \\\ -8 \ + \ (-9) & 13 \ + \ 4 \end{bmatrix} \ = \ \begin{bmatrix} 12 & -7 \\\ -17 & 17 \end{bmatrix}}\)

4) Find the answer: \(\begin{bmatrix} 24 & 31 \\\ -29 & 25 \end{bmatrix} + \begin{bmatrix} -17 & -26 \\\ 14 & 9 \end{bmatrix}\)

\(\color{red}{\begin{bmatrix} 24 & 31 \\\ -29 & 25 \end{bmatrix} + \begin{bmatrix} -17 & -26 \\\ 14 & 9 \end{bmatrix} \ = \ \begin{bmatrix} 24 \ + \ (-17) & 31 \ + \ (-26) \\\ -29 \ + \ 14 & 25 \ + \ 9 \end{bmatrix} \ = \ \begin{bmatrix} 7 & 5 \\\ -15 & 34 \end{bmatrix}}\)

5) Find the answer: \(\begin{bmatrix} 9 & 11 \\\ -14 & 5 \\\ 15 & 21 \end{bmatrix} + \begin{bmatrix} -18 & -7 \\\ 14 & -18 \\\ -29 & 3 \end{bmatrix}\)

\(\color{red}{\begin{bmatrix} 9 & 11 \\\ -14 & 5 \\\ 15 & 21 \end{bmatrix} + \begin{bmatrix} -18 & -7 \\\ 14 & -18 \\\ -29 & 3 \end{bmatrix} \ = \ \begin{bmatrix} 9 \ + \ (-18) & 11 \ + \ (-7) \\\ -14 \ + \ 14 & 5 \ + \ (-18) \\\ 15 \ + \ (-29) & 21 \ + \ 3 \end{bmatrix} \ = \ \begin{bmatrix} -9 & 4 \\\ 0 & -13 \\\ -14 & 24 \end{bmatrix}}\)

6) Find the answer: \(\begin{bmatrix} 2 & -5 \\\ 7 & 3 \end{bmatrix} - \begin{bmatrix} -4 & -2 \\\ 4 & -9 \end{bmatrix}\)

\(\color{red}{\begin{bmatrix} 2 & -5 \\\ 7 & 3 \end{bmatrix} + \begin{bmatrix} -4 & -2 \\\ 4 & -9 \end{bmatrix} \ = \ \begin{bmatrix} 2 \ - \ (-4) & -5 \ - \ (-2) \\\ 7 \ - \ 4 & 3 \ - \ (-9) \end{bmatrix} \ = \ \begin{bmatrix} 6 & -3 \\\ 3 & 12 \end{bmatrix}}\)

7) Find the answer: \(\begin{bmatrix} 17 & -28 \\\ -15 & 32 \end{bmatrix} - \begin{bmatrix} 12 & -5 \\\ 9 & -11 \end{bmatrix}\)

\(\color{red}{\begin{bmatrix} 17 & -28 \\\ -15 & 32 \end{bmatrix} - \begin{bmatrix} 12 & -5 \\\ 9 & -11 \end{bmatrix} \ = \ \begin{bmatrix} 17 \ - \ 12 & -28 \ - \ (-5) \\\ -15 \ - \ 9 & 32 \ - \ (-11) \end{bmatrix} \ = \ \begin{bmatrix} 5 & -23 \\\ -24 & 43 \end{bmatrix}}\)

8) Find the answer: \(\begin{bmatrix} 9 & -7 \\\ 18 & -19 \end{bmatrix} - \begin{bmatrix} -3 & 9 \\\ 6 & 4 \end{bmatrix}\)

\(\color{red}{\begin{bmatrix} 9 & -7 \\\ 18 & -19 \end{bmatrix} - \begin{bmatrix} -3 & 9 \\\ 6 & 4 \end{bmatrix} \ = \ \begin{bmatrix} 9 \ - \ (-3) & -7 \ - \ 9 \\\ 18 \ - \ 6 & -19 \ - \ 4 \end{bmatrix} \ = \ \begin{bmatrix} 12 & -16 \\\ 12 & -23 \end{bmatrix}}\)

9) Find the answer: \(\begin{bmatrix} 11 & 19 \\\ 27 & 34 \end{bmatrix} - \begin{bmatrix} -42 & -7 \\\ 5 & 39 \end{bmatrix}\)

\(\color{red}{\begin{bmatrix} 11 & 19 \\\ 27 & 34 \end{bmatrix} - \begin{bmatrix} -42 & -7 \\\ 5 & 39 \end{bmatrix} \ = \ \begin{bmatrix} 11 \ - \ (-42) & 19 \ - \ (-7) \\\ 27 \ - \ 5 & 34 \ - \ 39 \end{bmatrix} \ = \ \begin{bmatrix} 53 & 26 \\\ 22 & -5 \end{bmatrix}}\)

10) Find the answer: \(\begin{bmatrix} 18 & -14 \\\ 53 & -34 \\\ -13 & -15 \end{bmatrix} - \begin{bmatrix} 91 & -47 \\\ 19 & -20 \\\ 30 & -16 \end{bmatrix}\)

\(\color{red}{\begin{bmatrix} 18 & -14 \\\ 53 & -34 \\\ -13 & -15 \end{bmatrix} - \begin{bmatrix} 91 & -47 \\\ 19 & -20 \\\ 30 & -16 \end{bmatrix} \ = \ \begin{bmatrix} 18 \ - \ 91 & -14 \ - \ (-47) \\\ 53 \ - \ 19 & -34 \ - \ (-20) \\\ -13 \ - \ 30 & -15 \ - \ (-16) \end{bmatrix} \ = \ \begin{bmatrix} -73 & 33 \\\ 34 & -14 \\\ -43 & 1 \end{bmatrix}}\)

How to Add and Subtract Matrices