Question

Examine the consistency of the system of equations

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

Answer 1

The given system can be written as AX=B, where
$A=\left[\begin{array}{ccc}5& -1& 4\\ 2& 3& 5\\ 5& -2& 6\end{array}\right],X=\left[\begin{array}{c}x\\ y\\ z\end{array}\right]andB=\left[\begin{array}{c}5\\ 2\\ -1\end{array}\right]$
Here, $|A|=\left[\begin{array}{ccc}5& -1& 4\\ 2& 3& 5\\ 5& -2& 6\end{array}\right]=5(18+10)-(-1)(12-25)+4(-4-15)=140-13-76=51\ne 0$
$\therefore$ A is non -singular, Therefore, $A^{-1}$ exists.
Hence, the given system of equations is consistent.