Find the products AB and BA for the diagonal matrices. A=begin{bmatrix}3 & 0 &0 0 & -5&0 0&0&0 end{bmatrix}, B=begin{bmatrix}-7 & 0 &0 0 &4&0 0&0&12 end{bmatrix}

Question
Matrices
Find the products AB and BA for the diagonal matrices.
$$A=\begin{bmatrix}3 & 0 &0\\ 0 & -5&0 \\ 0&0&0 \end{bmatrix}, B=\begin{bmatrix}-7 & 0 &0\\ 0 &4&0 \\ 0&0&12 \end{bmatrix}$$

2021-01-03
Step 1
Given the diagonal matrix:
$$A=\begin{bmatrix}3 & 0 &0\\ 0 & -5&0 \\ 0&0&0 \end{bmatrix}, B=\begin{bmatrix}-7 & 0 &0\\ 0 &4&0 \\ 0&0&12 \end{bmatrix}$$
Step 2
Multiply the given matrix:
$$AB=\begin{bmatrix}3 & 0 &0\\ 0 & -5&0 \\ 0&0&0 \end{bmatrix}\begin{bmatrix}-7 & 0 &0\\ 0 &4&0 \\ 0&0&12 \end{bmatrix}$$
$$=\begin{bmatrix}3(-7) + 0 \cdot 0+0\cdot0&3\cdot0+0\cdot4+0\cdot0 & 3\cdot0+0\cdot0+0\cdot12 \\ 0\cdot(-7)+(-5)\cdot0+0\cdot0 &0\cdot0+(-5)\cdot4+0\cdot0 & 0\cdot0+(-5)\cdot0+0\cdot12 \\ 0\cdot(-7)+0\cdot0+0\cdot0 &0\cdot0+0\cdot4+0\cdot0&0\cdot0+0\cdot0+0\cdot12 \end{bmatrix}$$

$$=\begin{bmatrix}-21 & 0 &0\\0&-20&0\\ 0&0&0 \end{bmatrix}$$ Since the multiplication of diagonal matrices is commutative.
That is
If A and B are diagonal matrices, then AB = BA.
Thus,
$$AB=BA=\begin{bmatrix}-21 & 0 &0\\0&-20&0\\ 0&0&0 \end{bmatrix}$$

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