Matrices :Solve equations using the resultant product by finding a relation between each variable

Question

My Approach:

I have found the value of AB=8I(subscript 3) but I am confused about using this relation to solve the linear equations.

Any help would be appreciated.

Answer 1

Since $AB=8\operatorname{Id}$ (I didn't check it), then $\frac18B=A^{-1}$. Therefore\begin{align}\left\{\begin{array}{l}x-2y+3z=6\\x+4y+z=12\\x-3y+2z=1\end{array}\right.&\iff A.\begin{pmatrix}x\\y\\z\end{pmatrix}=\begin{pmatrix}6\\12\\1\end{pmatrix}\\&\iff\begin{pmatrix}x\\y\\z\end{pmatrix}=\frac18B\begin{pmatrix}6\\12\\1\end{pmatrix}.\end{align}

Answer 2

The system you want to solve can be written as $Ax=b$, and the solutions is $x=A^{-1}b$.

The result $AB=8I$ can tell you $A^{-1}$.