Similar to the addition of algebraic expressions or vectors, commutative law allows the addition of matrices without dependence on order. If \( A, B \in M_{m \times n} ( \mathbb{R}) \), we have that:
\( A + B = B + A \)
Consider \( A = \begin{bmatrix} 2 & -5 \\ -1 & 3 \end{bmatrix} \) and \( B = \begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix} \) (Recall addition of matrices from Operations on Matrices.)
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\( A + B \) = \( \begin{bmatrix} 2 & -5 \\ -1 & 3 \end{bmatrix} + \begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix} \) = \( \begin{bmatrix} 2+3 & -5+5 \\ -1+1 & 3+2 \end{bmatrix} \) = \( \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix} \) |
\( B + A \) = \( \begin{bmatrix} 3 & 5 \\ 1 & 2 \end{bmatrix} + \begin{bmatrix} 2 & -5 \\ -1 & 3 \end{bmatrix} \) \( \begin{bmatrix} 3+2 & 5+(-5) \\ 1+(-1) & 2+3 \end{bmatrix} \) \( \begin{bmatrix} 5 & 0 \\ 0 & 5 \end{bmatrix} \) |
We see that \( A + B = B + A \)
When dealing with matrices, we can apply the associative laws of addition and multiplication. This means that if we have \( A, B, C \in M_{m \times n} (\mathbb{R}) \) then:
\( 1) \) \( A + (B + C) = (A + B) + C \)
\( 2) \) \( A(BC) = (AB)C \)
| Note: \( AB \neq BA \) and \( A(BC) \neq (BC)A \). Matrix multiplication is not commutative! |
If we are dealing with matrix multiplication combined with matrix addition, we can apply the distributive law. If \( A, B, C \in M_{m \times n} ( \mathbb{R}) \) then:
\( A(B + C) = AB + AC \)
