# How to Solve a System of Equations Using Matrices?

Any system of linear equations can be written as a matrix equation. In this step-by-step guide, you learn how to solve a system of equations using matrices.

## Step-by-step guide to solving a system of equations using matrices

When solving a system of equations using matrices, we have three matrices $$A, B$$, and $$X$$, where $$A$$ is known as the coefficient matrix, $$B$$ is the constant matrix, and $$X$$ contains all the variables of the equations, known as a variable matrix. Matrix $$A$$ is of the order $$m×n$$, while $$B$$ is the column matrix of the order $$m×1$$. The product of matrix $$A$$ and matrix $$X$$ reaches matrix $$B$$. Hence, $$X$$ is a column matrix of order $$n×1$$.

The matrices are arranged as:

$$\color{blue}{A . X = B}$$

Let’s understand how to solve a system of equations using a matrix with the help of an example. We have a set of two equations given below. The equations are:

$$\begin{cases}x+y=6 \\ 2x+3y=14\end{cases}$$

Arrange all the coefficients, variables, and constants of the matrix in such a way that whenever we find the product of the matrices, the obtained result should be an equation. Then the matrix equation is, $$AX = B$$ where:

$$A= \begin{bmatrix}1 & 1 \\2 & 3 \end{bmatrix}$$

$$X= \begin{bmatrix}x \\y \end{bmatrix}$$

$$B= \begin{bmatrix}6 \\14 \end{bmatrix}$$

We need to find matrix $$X$$, to solve the equations. It can be found by multiplying the inverse of matrix $$A$$ with $$B$$, which is obtained as $$X=\left(A^{-1}\right)B$$.

To find the determinant of matrix $$A$$, we follow the following steps:

$$|A|= \begin{bmatrix}1 & 1 \\2 & 3 \end{bmatrix}$$

Therefore, $$|A|= 3\: – 2 = 1$$

$$|A|≠0$$, it is possible to find the inverse of matrix $$A$$.

Now, by using the formula for finding the inverse of $$2×2$$ matrix:

$$A^{-1}= \begin{bmatrix}3 & -1 \\-2 & 1 \end{bmatrix}$$

Now to find the matrix $$X$$, we’ll multiply $$A^{-1}$$ and $$B$$. We get,

$$\begin{bmatrix}3 & -1 \\-2 & 1 \end{bmatrix}$$$$\begin{bmatrix}6 \\14 \end{bmatrix}$$$$=\begin{bmatrix}4\\2 \end{bmatrix}$$

So, the value of matrix $$X$$ is, $$\begin{bmatrix}4\\2 \end{bmatrix}$$.

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