Matrix Inversion Method

Matrix inversion method applies when the coefficient matrix $A$ of $AX=B$ is square and non-singular.

Since $A^{-1}$ exists, pre-multiply both sides of $AX=B$ by $A^{-1}$:

Worked illustration. Solve $5x+2y=3,\ 3x+2y=5$. Here $A=\begin{pmatrix}5&2\\3&2\end{pmatrix}$, $B=\begin{pmatrix}3\\5\end{pmatrix}$; $|A|=10-6=4\ne0$, so $A^{-1}$ exists: $A^{-1}=\frac14\begin{pmatrix}2&-2\\-3&5\end{pmatrix}$. Then $X=A^{-1}B=\frac14\begin{pmatrix}2&-2\\-3&5\end{pmatrix}\begin{pmatrix}3\\5\end{pmatrix}=\frac14\begin{pmatrix}6-10\\-9+25\end{pmatrix}=\frac14\begin{pmatrix}-4\\16\end{pmatrix}=\begin{pmatrix}-1\\4\end{pmatrix}$, i.e. $x=-1,y=4$ -- check: $5(-1)+2(4)=3$ and $3(-1)+2(4)=5$, both correct.

Practical recipe (order 3). Read $A$ and $B$ off the equations; compute $|A|$; compute $\operatorname{adj}A$ from the nine cofactors; form $A^{-1}=\frac1{|A|}\operatorname{adj}A$; multiply $A^{-1}B$ and read $x,y,z$ off the resulting column. …