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Mathematics Question on Determinants

Examine the consistency of the system of equations.
3x-y−2z=2,
2y−z=−1
3x−5y=3

Answer

The given system of equations is:
3x − y − 2z = 2
2y − z = −1
3x − 5y = 3
This system of equations can be written in the form of AX = B, where

A=[3−1−2\02−1\3−50]\begin{bmatrix}3&-1&-2\\\0&2&-1\\\3&-5&0\end{bmatrix}, X= [x\y\z]\begin{bmatrix}x\\\y\\\z\end{bmatrix} and B=[2−1\3]\begin{bmatrix}2\\\\-1\\\3\end{bmatrix}

Now IAI=3(0-5)-0+3(1+4)=-15+15=0
∴ A is a singular matrix.
Now (adj A)=[−5105−363−6126]\begin{bmatrix}-5&10&5\\\\-3&6&3\\\\-6&12&6\end{bmatrix}

so (adj A)B=\begin{bmatrix}-5&10&5\\\\-3&6&3\\\\-6&12&6\end{bmatrix}$$\begin{bmatrix}2\\\\-1\\\3\end{bmatrix}

=[−10−10+15−6−6+9−12−12+18]\begin{bmatrix}-10-10+15\\\\-6-6+9\\\\-12-12+18\end{bmatrix}=\begin{bmatrix}-5\\\\-3\\\\-6\end{bmatrix}$$\neq 0

Thus, the solution of the given system of equations does not exist. Hence, the system of
equations is inconsistent.