Matrix Inversion Method and Cramer's Rule

Matrix Inversion Method and Cramer's Rule

5x+4y=14, 3x+7y=13

This system of equations can be written in matrix equation as follows:

where $\mathbf{A} = \left[ \begin{array}{ll}5 & 4\\ 3 & 7 \end{array} \right]$ is a matrix of coefficients, $\mathbf{X} = \left[ \begin{array}{l}x\\ y \end{array} \right]$ is a matrix of variables and $\mathbf{B} = \left[ \begin{array}{l}14\\ 13 \end{array} \right]$ is a matrix of constants.

Now equation can also be written as:

This equation indicates that the values of the variables in the matrix $\mathbf{X}$ are equal to the corresponding elements in the matrix $\mathbf{A}^{-1}\mathbf{B}$.

As, $\left| \mathbf{A} \right| = 35 - 12 = 23 \neq 0$ , so inverse is possible.

adj $\mathbf{A} = \left[ \begin{array}{ll}7 & - 4\\ - 3 & 5 \end{array} \right]$