If X=pmatrixxpmatrix is a solution of the system of equations AX=B, where adj(A)=pmatrix4&…

Question

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

Accepted Answer

$X=A^{-1}B=\dfrac{\operatorname{adj}(A)}{|A|}B$.

$\operatorname{adj}(A)\cdot B=\begin{pmatrix}4&2&2\-5&0&5\1&-2&3\end{pmatrix}\begin{pmatrix}4\0\2\end{pmatrix}=\begin{pmatrix}20\-10\10\end{pmatrix}$.

For $X$ to have the simple form used in the solution, $|A|=\pm10$, giving $X=\pm\begin{pmatrix}2\-1\1\end{pmatrix}$.

So $x+y+z=\pm2$, and $|x+y+z|=2$.

Solutions · 1