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Mathematics Question on Determinants

Solve system of linear equations, using matrix method.
4x-3y=3
3x-5y=7

Answer

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

A=[4−3\3−5]\begin{bmatrix}4&-3\\\3&-5\end{bmatrix},X=[x\y]\begin{bmatrix}x\\\y\end{bmatrix}and B=[3\7]\begin{bmatrix}3\\\7\end{bmatrix}.

Now, |A|=-20+9=-11≠0
Thus, A is non-singular.
Therefore, its inverse exists.

Now,A-1=1∣A∣\frac{1}{\mid A\mid}adj A=-111\frac{1}{11}[5-334]=\frac{1}{11}$$\begin{bmatrix}5&-3\\\3&-4\end{bmatrix}

∴X=A-1 B=\frac{1}{11}$$\begin{bmatrix}5&-3\\\3&-4\end{bmatrix}$$\begin{bmatrix}3\\\7\end{bmatrix}

\Rightarrow \begin{bmatrix}x\\\y\end{bmatrix}=\frac{1}{11}$$\begin{bmatrix}5&-3\\\3&-4\end{bmatrix}$$\begin{bmatrix}3\\\7\end{bmatrix}=\frac{1}{11}$$\begin{bmatrix}15-21\\\9-28\end{bmatrix}

=\frac{1}{11}$$\begin{bmatrix}-6\\\\-19\end{bmatrix}=[−611−1911]\begin{bmatrix}-\frac{6}{11}\\\\-\frac{19}{11}\end{bmatrix}

Hence, x=−611 and y=−1911-\frac{6}{11}\,and \,y=-\frac{19}{11}