[tex]-3x +2y -6z = 5\\-7x + 9y -1z = 4\\-2x +5y +2z = 9[/tex]
[tex]\left[\begin{array}{ccc}-3&2&-6\\-7&9&-1\\-2&5&2\end{array}\right] \left[\begin{array}{c}x\\y\\z\end{array}\right] = \left[\begin{array}{c}5\\4\\9\end{array}\right][/tex]
Let
[tex]A = \left[\begin{array}{ccc}-3&2&-6\\-7&9&-1\\-2&5&2\end{array}\right][/tex]
[tex]det(A) = -3det\left( \left[\begin{array}{cc}9&-1\\5 & 2\end{array}\right] \right) -2 det\left( \left[\begin{array}{cc}-7&-1\\-2 & 2\end{array}\right] \right) -6 det\left( \left[\begin{array}{cc}-7&9\\-2 & 5\end{array}\right] \right)[/tex]
[tex]det(A) = -3(18+5) -2(-14-2) -6(-35+18) = -3(23) -2(-16) -6(-17)\\= -69 +32 +102= 65[/tex]
