Consider system of equations in x and y
2x+3y=5
and 3x+4y=7
then x and y can ve evaluated by matrix method
Solution-
A=\begin{bmatrix} 2 &3 \\3 &4 \\ \end{bmatrix}
Matrix X can be written as \[\begin{bmatrix} x\\ y \\ \end{bmatrix}\]
Matrix B can be written as \[\begin{bmatrix} 5\\ 7 \\ \end{bmatrix}\]
then x and y can be evaluated as X=\[A^{-1}\]B where \[A^{-1}\] represents inverse of a matrix .
\[A^{-1}=\frac{adjA}{detA}\]
Adjoint of matrix is found out by taking transpose of cofactor matrix.
A=\begin{bmatrix} 2 &3 \\3 &4 \\ \end{bmatrix}\
Minor of the matrix A is \[ \begin{bmatrix} 4 & 3 \\ 3& 2 \\ \end{bmatrix}\]
cofactor of A=\[ \begin{bmatrix} 4 & -3 \\ -3& 2 \\ \end{bmatrix}\]
transpose of the cofactor matrix is \[ \begin{bmatrix} 4 & -3 \\ -3& 2 \\ \end{bmatrix}\]
determinant of Matrix A=-1
X=\[\frac{\begin{bmatrix} 4 &-3 \\-3 &2 \\ \end{bmatrix}}{-1}\]\[ \begin{bmatrix}5 \\ 7\\ \end{bmatrix}\]=\[ \begin{bmatrix}1 \\ 1\\ \end{bmatrix}\]
