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If A = [2513], B =[1−1−32],\begin{bmatrix}[r] 2 & 5 \ 1 & 3 \end{bmatrix}, \text{ B } = \begin{bmatrix}[r] 1 & -1 \ -3 & 2 \end{bmatrix}, find AB and BA. Is AB = BA ?

Matrices

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Answer

 AB =[2513][1−1−32]=[2.1+5.(−3)2.(−1)+5.21.1+3.(−3)1.(−1)+3.2]=[−138−85] BA =[1−1−32][2513]=[1×2+(−1)×11×5+(−1)×3−3×2+2×1−3×5+2×3]=[2−15−3−6+2−15+6]=[12−4−9]\text{ AB } = \begin{bmatrix}[r] 2 & 5 \ 1 & 3 \end{bmatrix} \begin{bmatrix}[r] 1 & -1 \ -3 & 2 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 2.1 + 5.(-3) & 2.(-1) + 5.2 \ 1.1 + 3.(-3) & 1.(-1) + 3.2 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] -13 & 8 \ -8 & 5 \end{bmatrix} \\[1em] \text{ BA } = \begin{bmatrix}[r] 1 & -1 \ -3 & 2 \end{bmatrix} \begin{bmatrix}[r] 2 & 5 \ 1 & 3 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 1 \times 2 + (-1) \times 1 & 1 \times 5 + (-1) \times 3 \ -3 \times 2 + 2 \times 1 & -3 \times 5 + 2 \times 3 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 2 - 1 & 5 - 3 \ -6 + 2 & -15 + 6 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 1 & 2 \ -4 & -9 \end{bmatrix}

The matrix AB = [−138−85] and BA =[12−4−9].\begin{bmatrix}[r] -13 & 8 \ -8 & 5 \end{bmatrix} \text{ and BA } = \begin{bmatrix}[r] 1 & 2 \ -4 & -9 \end{bmatrix}. AB ≠ BA.

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