##### Education

# Matrix Method of Solving Linear Equations

Linear equations are algebraic equations whose degree of each variable is 1. Graphically, linear equations represent straight lines in a plane. To solve a system of linear equations, we have a number of methods.

We can solve them algebraically, use matrices and may use iterative methods. In this article, we shall discuss a matrix method – Cramers rule to find the solution to the given system of linear equations.

A system of linear equations

a11x1 + a12x2 + … + a1nxn = b1

a21x1 + a22x2 + … + a2nxn = b2

⁞

am1x1 + am2x2 + … + amnxn = bm

can be represented in matrix form as AX = B, where

This system of linear equations can be easily solved using Cramer’s Rule.

## Cramer’s Rule

This rule is named after the mathematician Gabriel Cramer (1704–1752). In this method, the system of linear equations is solved by taking the determinants of the matrices. Let AX = B represent a system of linear equations, where A is the (m × n) coefficient matrix, X is the (n × 1) column matrix of unknowns, and B is the (m × 1) column matrix of constants.

To get the solution by this rule, we first have to find the determinants D, Dx1, Dx2, …, Dxn where

D = |A|; determinant of matrix A,

[latex]D=\begin{vmatrix}a_{11} &a_{12} & … & a_{1n} \\a_{21} &a_{22} & … & a_{2n} \\ …&… &… & … \\a_{m1} &a_{m2} & … & a_{mn} \\\end{vmatrix}[/latex]

Dx1 = The determinant of matrix A whose 1st column is replaced by the column matrix B,

[latex]Dx_{1}=\begin{vmatrix}b_{1} &a_{12} & … & a_{1n} \\b_{2} &a_{22} & … & a_{2n} \\ …&… &… & … \\b_{m} &a_{m2} & … & a_{mn} \\\end{vmatrix}[/latex]

Dx2 = The determinant of matrix A whose 2nd column is replaced by the column matrix B,

[latex]Dx_{2}=\begin{vmatrix}a_{11} &b_{1} & … & a_{1n} \\a_{21} &b_{2} & … & a_{2n} \\ …&… &… & … \\a_{m1} &b_{m} & … & a_{mn} \\\end{vmatrix}[/latex]

⁞

Dxn = The determinant of matrix A whose nth column is replaced by the column matrix B.

[latex]Dx_{n}=\begin{vmatrix}a_{11} &a_{12} & … & b_{1} \\a_{21} &a_{22} & … & b_{2} \\ …&… &… & … \\a_{m1} &a_{m2} & … & b_{m} \\\end{vmatrix}[/latex]

Then, x1 = Dx1/D, x2 = Dx2/D, …, xn = Dxn/D

Also, refer to Cramer’s Rule in detail.

### Important Points to Remember

- Cramer’s Rule is applicable only if D ≠ 0; that is, the coefficient of matrix A should not be a singular matrix.

- If D ≠ 0, the system of equations AX = B has a unique solution.
- If D = 0, then the given linear equation system either has no solution or infinitely many solutions.

## Steps to Apply Cramer’s Rule

Let us understand how to apply Cramer’s rule with an example. Consider the system of linear equations: 5x + 7y = – 2 and 4x + 6y = – 3

**Step 1: **Express in a matrix form of AX = B

Thus,

[latex]A=\begin{bmatrix}5 & 7 \\4 & 6 \\\end{bmatrix}, \:\;X=\begin{bmatrix}x \\y\end{bmatrix}\:\;and\:\;B=\begin{bmatrix}-2 \\-3\end{bmatrix} [/latex]

**Step 2: ** Now, we shall find the determinants.

Find the subtraction of a12a21 from a11a22 to get the value of D.

[latex]D=|A|=\begin{vmatrix}5 & 7 \\4 & 6 \\\end{vmatrix}= 5\times 6 – 4 \times 7=30-28=2[/latex]

Find the subtraction of a12b2 from a22b1 by replacing column 1 of A with B to get the value of Dx.

[latex]Dx=\begin{vmatrix}-2 & 7 \\-3 & 6 \\\end{vmatrix}= -2\times 6 – (-3) \times 7=-12+21=9[/latex]

Finally, find the subtraction of a12b2 from a22b1 by replacing column 2 of A with B to get the value of Dy.

[latex]Dy=\begin{vmatrix}5 & -2 \\4 & -3 \\\end{vmatrix}= 5\times (-3) – (-2) \times 4=-15+8=-7[/latex]

**Step 3: **Find the value of unknowns.

Now x = Dx/D = 9/2 and y = Dy/D = –7/2

Thus, the solution of the given system of linear equations by Cramer’s Rule is x = 9/2 and y = –7/2.

This rule is efficient in finding solutions for systems of linear equations of two or three variables. For more number of variables, this method becomes quite tedious.