How to find the solution of a system of linear equations using matrices?

To find the solution of a system of linear equations using matrices, we can use matrix algebra.

First, we need to write the system of equations in matrix form. For example, consider the system:

2x + 3y = 7
4x - 5y = -1

We can write this as the matrix equation:

\begin{bmatrix} 2 & 3 \\ 4 & -5 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 7 \\ -1 \end{bmatrix}

Next, we can use matrix algebra to solve for the variables. We can start by finding the inverse of the coefficient matrix:

\begin{bmatrix} 2 & 3 \\ 4 & -5 \end{bmatrix}^{-1} = \frac{1}{2\times(-5)-3\times4} \begin{bmatrix} -5 & -3 \\ -4 & 2 \end{bmatrix} = \begin{bmatrix} \frac{5}{22} & \frac{3}{22} \\ \frac{2}{11} & -\frac{1}{11} \end{bmatrix}

Then, we can multiply both sides of the matrix equation by the inverse of the coefficient matrix:

\begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} \frac{5}{22} & \frac{3}{22} \\ \frac{2}{11} & -\frac{1}{11} \end{bmatrix} \begin{bmatrix} 7 \\ -1 \end{bmatrix} = \begin{bmatrix} 1 \\ 2 \end{bmatrix}

Therefore, the solution to the system of equations is x = 1 and y = 2. We can check this by substituting these values back into the original equations and verifying that they are true.