Question

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

Answer 1

The given system of equations can be written in the form of AX = B, where
A=\(\begin{vmatrix} 2 &-1 \\   3& 4 \end{vmatrix}\), X=\(\begin{vmatrix} x  \\  y  \end{vmatrix}\)and B=\(\begin{vmatrix} -2  \\   3  \end{vmatrix}\)
\(Now |A|=8+3=11≠0\)
Thus, A is non-singular. Therefore, its inverse exists.
Now,
 A^-1=\(\frac{1}{|A|}\)adjA=\(\frac{1}{11}\)\(\begin{vmatrix} 4 &1 \\   -3& 2 \end{vmatrix}\)
so X=A^-1B=\(\frac{1}{11}\)\(\begin{vmatrix} 4 &1 \\   -3& 2 \end{vmatrix}\)\(\begin{vmatrix} -2  \\   3  \end{vmatrix}\)
\(\Rightarrow \begin{vmatrix} x \\   y  \end{vmatrix}=\frac{1}{11}\begin{vmatrix}  -8+3 \\    6+6 \end{vmatrix}=\frac{1}{11}\begin{vmatrix} -5  \\   12  \end{vmatrix}\)=\(\begin{vmatrix} -\frac{5}{11}  \\   \frac{12}{11}  \end{vmatrix}\)
Hence x=\(-\frac{5}{11}\) and y=\(\frac{12}{11}\)