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If matrix A = [1321]\begin{bmatrix}[r] 1 & 3 \ 2 & 1 \end{bmatrix} and matrix B = [4−1]\begin{bmatrix}[r] 4 \ -1 \end{bmatrix}, find : AB - B.

Matrices

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Answer

Solving,

⇒AB−B=[1321][4−1]−[4−1]=[1×4+3×−12×4+1×−1]−[4−1]=[4−38−1]−[4−1]=[17]−[4−1]=[1−47−(−1)]=[−38].\Rightarrow AB - B = \begin{bmatrix}[r] 1 & 3 \ 2 & 1 \end{bmatrix}\begin{bmatrix}[r] 4 \ -1 \end{bmatrix} - \begin{bmatrix}[r] 4 \ -1 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 1 \times 4 + 3 \times -1 \ 2 \times 4 + 1 \times -1 \end{bmatrix} - \begin{bmatrix}[r] 4 \ -1 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 4 - 3 \ 8 - 1 \end{bmatrix} - \begin{bmatrix}[r] 4 \ -1 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 1 \ 7 \end{bmatrix} - \begin{bmatrix}[r] 4 \ -1 \end{bmatrix} \\[1em] = \begin{bmatrix}[r] 1 - 4 \ 7 - (-1) \end{bmatrix} \\[1em] = \begin{bmatrix}[r] -3 \ 8 \end{bmatrix}.

Hence, AB - B = [−38].\begin{bmatrix}[r] -3 \ 8 \end{bmatrix}.

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