If $X = \begin{bmatrix} x \ y \ z \end{bmatrix}$ is a solution of the system of equations$AX = B$,

Question

If $X = \begin{bmatrix} x \ y \ z \end{bmatrix}$ is a solution of the system of equations$AX = B$, where adj $A = \begin{bmatrix} 4 & 2 & 2 \ - 5 & 0 & 5 \ 1 & - 2 & 3 \end{bmatrix}$ and$B = \begin{bmatrix} 4 \ 0 \ 2 \end{bmatrix}$, then $|x + y + z|$ is equal to :

Accepted Answer

$\ X = A^{- 1}B = \left( \frac{adjA}{|A|} \right)B$

$${= \pm \frac{1}{10}\begin{pmatrix} 4 & 2 & 2 \\ - 5 & 0 & 5 \\ 1 & - 2 & 3 \end{pmatrix}\begin{pmatrix} 4 \\ 0 \\ 2 \end{pmatrix} }{= \pm \frac{1}{10}\begin{pmatrix} 20 \\ - 10 \\ 10 \end{pmatrix} = \pm \begin{pmatrix} 2 \\ - 1 \\ 1 \end{pmatrix} }{\therefore|x + y + z| = 2}$$

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