# Examine the consistency of the system of equations: $\quad 5x -y +4z = 5$ $\quad2x + 3y + 5z= 2$ $\quad 5x - 2y + 6z = -1$

Toolbox:
• (i)A matrix is said to be singular if |A|$\neq 0$.
• (ii)If A is a non-singular matrix such that
• AX=B.
• then $X=A^{-1}X$
• Using this we can solve the system of equation which has unique solution.
This can be written in the form AX=B.

$\begin{bmatrix}5 & -1& 4\\2 & 3& 5\\5 & -2 &6\end{bmatrix}\begin{bmatrix}x\\y\\z\end{bmatrix}=\begin{bmatrix}5\\2\\1\end{bmatrix}$

Where $A=\begin{bmatrix}5 & -1& 4\\2 & 3& 5\\5 & -2&6\end{bmatrix}\;X=\begin{bmatrix}x\\y\\z\end{bmatrix}\;B=\begin{bmatrix}5\\2\\1\end{bmatrix}$

Let us now calculate the determinant value of A by expanding along $R_1$

|A|=$5(3\times 6-5\times -2)-(-1)(2\times 6-5\times 5)+4(2\times -2-5\times 3)$

$\;\;\;=5(18+10)+1(12-25)+4(-4-15).$

$\;\;\;=5(28)+1(-13)+4(-19).$

$\;\;\;=140-13-76=51\neq 0.$

Therefore A is non-singular.

Hence inverse exists.

Hence the given system of equation is consistent.

edited Feb 27, 2013