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If A = [−1324] and B =[2−3−4−6],\begin{bmatrix}[r] -1 & 3 \ 2 & 4 \end{bmatrix} \text{ and B } = \begin{bmatrix}[r] 2 & -3 \ -4 & -6 \end{bmatrix}, find the matrix AB + BA.

Matrices

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Answer

 AB =[−1324][2−3−4−6]=[−1×2+3×−4−1×(−3)+3×(−6)2×2+4×(−4)2×(−3)+4×(−6)]=[−2−123−184−16−6−24]=[−14−15−12−30]BA =[2−3−4−6][−1324]=[2×(−1)+(−3)×22×3+(−3)×4(−4)×(−1)+(−6)×2(−4)×3+(−6)×4]=[−2−66−124−12−12−24]=[−8−6−8−36]Given, AB + BA =[−14−15−12−30]+[−8−6−8−36]=[−14+(−8)−15+(−6)−12+(−8)−30+(−36)]=[−22−21−20−66].\text{ AB } = \begin{bmatrix}[r] -1 & 3 \ 2 & 4 \end{bmatrix} \begin{bmatrix}[r] 2 & -3 \ -4 & -6 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] -1 \times 2 + 3 \times -4 & -1 \times (-3) + 3 \times (-6) \ 2 \times 2 + 4 \times (-4) & 2 \times (-3) + 4 \times (-6) \end{bmatrix} \\[1em] = \begin{bmatrix}[r] -2 - 12 & 3 - 18 \ 4 - 16 & -6 - 24 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] -14 & -15 \ -12 & -30 \end{bmatrix} \\[1em] \text{BA } = \begin{bmatrix}[r] 2 & -3 \ -4 & -6 \end{bmatrix} \begin{bmatrix}[r] -1 & 3 \ 2 & 4 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 2 \times (-1) + (-3) \times 2 & 2 \times 3 + (-3) \times 4 \ (-4) \times (-1) + (-6) \times 2 & (-4) \times 3 + (-6) \times 4 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] -2 - 6 & 6 - 12 \ 4 - 12 & -12 - 24 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] -8 & -6 \ -8 & -36 \end{bmatrix} \\[1em] \text{Given, AB + BA } = \begin{bmatrix}[r] -14 & -15 \ -12 & -30 \end{bmatrix} + \begin{bmatrix}[r] -8 & -6 \ -8 & -36 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] -14 + (-8) & -15 + (-6) \ -12 + (-8) & -30 + (-36) \end{bmatrix} \\[1em] = \begin{bmatrix}[r] -22 & -21 \ -20 & -66 \end{bmatrix}.

Hence, the matrix AB + BA = [−22−21−20−66].\begin{bmatrix}[r] -22 & -21 \ -20 & -66 \end{bmatrix}.

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