$5x - y + 4z = 5,2x + 3y + 5z = 2,5x - 2y + 6z = - 1.$

The system of equations can be written in the form$AX = B$ ,

where
$A = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}5&{ - 1}&4\\2&3&5\\5&{ - 2}&6\end{array}} \right],X = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}x\\y\\z\end{array}} \right]and\;B = \left[ {\begin{array}{rrrrrrrrrrrrrrrrrrrr}5\\2\\{ - 1}\end{array}} \right]$

Now, $|A| = \left| {\begin{array}{rrrrrrrrrrrrrrrrrrrr}5&{ - 1}&4\\2&3&5\\5&{ - 2}&5\end{array}} \right| = 5(18 + 10) + 11(12 - 25) + 4( - 4 - 15)$

$= 140 - 13 - 76 = 51 \ne 0$

Hence, equations are consistent with a unique solution.

Solve system of linear equations, using matrix method, in questions 7 to 14.