Q.If $x = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$, $B = \begin{bmatrix} 8 \\ 1 \\ 4 \end{bmatrix}$, $A^{-1} = \frac{1}{17}\begin{bmatrix} -1 & -5 & -1 \\ -8 & -6 & 9 \\ -10 & 1 & 7 \end{bmatrix}$ and $x = A^{-1} \cdot B$, then find the values of $x$, $y$ and $z$.

Multiply $A^{-1}$ by $B$ row-wise and divide by 17.

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$x=A^{-1}B=\tfrac{1}{17}(-17,-34,-51)^T=(-1,-2,-3)^T$.

We are given

$A^{-1}=\frac{1}{17}\begin{bmatrix}-1&-5&-1\\-8&-6&9\\-10&1&7\end{bmatrix},\qquad B=\begin{bmatrix}8\\1\\4\end{bmatrix},\qquad x=A^{-1}B.$

Multiply the matrix $A^{-1}$ by the column $B$, row by row (before applying the factor $\tfrac{1}{17}$):

$\text{Row 1: } (-1)(8)+(-5)(1)+(-1)(4)=-8-5-4=-17,$

$\text{Row 2: } (-8)(8)+(-6)(1)+(9)(4)=-64-6+36=-34,$ …