Cramer's rule for system of linear equations

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akhl

@akhl

Dazzler

Posted: 15 years ago

Cramer's Rule

Cramer's rule can be used to solve a system of linear equations if a solution exists. It can also tell if there is a unique solution set, no solution, or infinitely many solution sets.

Consider the linear system of equations:

ax + by = e

cx + dy = f

It can be written in matrix form as

So, write the coefficients in the first matrix called coefficient matrix. The coefficients appear in the same position in the matrix as in the given equations. Multiply it with a column vector which contains the variables. On the rhs, there is results column vector containing the rhs of the given equations.

To solve the equations using Cramer's rule find the following determinants:

There is a simple way to remember these determinants. The determinant D is the determinant of the coefficient matrix. To get the determinant of Dx, replace the elements of x column in D by the elements of the result column vector. To get the determinant of Dy, replace the elements of y column in D by the elements of the result column vector. The elements in the result column vector are e ad f. The variable x is the first variable. Therefore in Dx, the first column contains e and f. The variable y is the second variable. Therefore in Dy, the second column contains e and f. In general, you can find the determinant for any variable by replacing the column for that variable in the determinant of the coefficient matrix by the elements in the result column vector.

The rules for 3 equations in 3 variables are the same. Consider

ax + by + cz = j

dx + ey + fz = k

gx + hy + iz = l

The above set of equations can be written in matrix form as

Find the following determinants:-

As explained in the case of equations in two variables, D is the determinant of the coefficient matrix. Dx is obtained by replacing x-column in D by the elements of the result column. Likewise, Dy is obtained by replacing y-column in D by the elements of the result column, and Dz is obtained by replacing z-column in D by the elements of the result column.

Let us consider different cases:-

1. All of the determinants are non-zero: There is a unique solution set, i.e., there is exactly one set of values of the variables which will satisfy the given equations. Find the solution using the formulas:

x = Dx/D, y = Dy/D

For three variables, z = Dz/D

2. All of the determinants are zero: There are infinitely many solution sets, i.e., more than one set of the values of the variables will satisfy the given equations.

When there are infinitely many solution sets, then the equations are said to be indeterminate.

3. D = 0 and at least one of the other determinants is not zero: There is no solution set, i.e., no values of the variables will satisfy all the equations.

When there is no solution set, then the equations are said to be inconsistent or incompatible.

The steps explained in this tutorial can be extended to 4 or more variables.

Edited by akhl - 15 years ago