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Question

Mathematics Question on Determinants

Solve system of linear equations, using matrix method.
5x+2y=4,
7x+3y=5

Answer

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

A=[52\73]\begin{bmatrix}5&2\\\7&3\end{bmatrix},X=[x\y]\begin{bmatrix}x\\\y\end{bmatrix} and B=[4\5]\begin{bmatrix}4\\\5\end{bmatrix}

Now IAI=15-14=1 ≠0
Thus, A is non-singular. Therefore, its inverse exists.
Now A-1=1∣A∣\frac{1}{\mid A\mid}adj A

A-1=[3−2−75]\begin{bmatrix}3&-2\\\\-7&5\end{bmatrix}

X=A-1B=\begin{bmatrix}3&-2\\\\-7&5\end{bmatrix}$$\begin{bmatrix}4\\\5\end{bmatrix}

⇒[x\y]\Rightarrow \begin{bmatrix}x\\\y\end{bmatrix}=[−12−10−28+25]\begin{bmatrix}-12-10\\\\-28+25\end{bmatrix}=[2−3]\begin{bmatrix}2\\\\-3\end{bmatrix}

Hence x=2 and Y=-3.