Q.Solve the system $2x-y+z=3$, $-x+2y-z=0$, $x-y+2z=5$ by the matrix inversion method.

With three equations, the same $X=A^{-1}B$ formula applies, using a $3\times3$ coefficient matrix and its inverse found by the adjoint method.

Step 1 — Write the system as $AX=B$

Step 2 — Use $A^{-1}$

This is the same coefficient matrix as the earlier exercise on the inverse of a $3\times3$ matrix, where $|A|=4\neq0$ and

Step 3 — Compute $X=A^{-1}B$

$\text{adj}(A)\cdot B=\begin{pmatrix}3(3)+1(0)+(-1)(5)\\1(3)+3(0)+1(5)\\-1(3)+1(0)+3(5)\end{pmatrix}=\begin{pmatrix}9-5\\3+5\\-3+15\end{pmatrix}=\begin{pmatrix}4\\8\\12\end{pmatrix}$ …

Making an arithmetic slip in the three-row matrix-vector multiplication $\text{adj}(A)\cdot B$ (mixing up which row of the adjoint pairs with $B$) — always keep the row inde …