example of Cramer’s rule

Say we want to solve the system

$3x+2y+z-2w$$=$$4$

$2x-y+2z-5w$$=$$15$

$4x+2y-5w$$=$$1$

$3x-2z-4w$$=$$1.$

The associated matrix is

whose determinant is $\mathrm{\Delta }=-65$. Since the determinant is non-zero, we can use Cramer’s rule. To obtain the value of the $k$-th variable, we replace the $k$-th column of the matrix above by the column vector

$$\left(\begin{array}{c}\hfill 4\hfill \\ \hfill 15\hfill \\ \hfill 1\hfill \\ \hfill 1\hfill \end{array}\right),$$

the determinant of the obtained matrix is divided by $\mathrm{\Delta }$ and the resulting value is the wanted solution.

So