Matrix: Evaluating Matrices using Crammer’s Rule

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In mathematics, Cramer’s rule is used to solve the simultaneous linear equation. It is widely used in algebra. Cramer’s rule is a type of matrix. Matrix is a rectangular array of numbers enclosed in “[]” brackets. 

In this post, we will learn all the basic concepts of Cramer’s rule along with a lot of examples. We shall also discuss determinants.

What is Cramer’s Rule?

Cramer’s rule is a way of locating values of variables of linear equations with matrices. In linear algebra, Cramer’s rule is an explicit method for the solution of a system of linear equations with as many equations as known. It is also known as a determinant method. Cramer’s rule makes use of determinants to solve for a solution to the equation.

 In Cramer’s rule, the equation or matrix should be a square matrix or the equal number of rows and columns as Cramer’s rule is not applicable for the rectangular array. The question will raise in mind that the definition of matrix tills that the matrix is a rectangular array of numbers and the Cramer’s rule is not applicable on rectangular array although it is a type of matrix? 

The answer is every square matric is a rectangular matrix. So, in Cramer’s rule, we are dealing with a rectangular array but the numbers of rows will be equal to the number of columns and when the number of rows and the number of columns is same then we said this to be a square matrix.

The form of Cramer’s rule is 

Ax = b

A is a square matrix, x to be found, and b is the result. Ax = b is called matrix equation. Can be written as,

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A1x + B1y = C1

A2x + B2y = C2

By interchanging the values, we will make matrices to find x and y.

After making the matrices, we have to find the determinant of the square matrix given in the problem, x-matrix, and y-matrix. After finding the determinants keep one thing in mind, if the determinant of the given matrix is zero, we cannot apply Cramer’s rule. So, if the determinant is non-zero, then we apply Cramer’s rule to find the value of x and y by applying the formula.

Value of x,

X = det[x]/det 

Value of y,

y = det[y]/det 

How to use Cramer’s Rule to evaluate matrices?

In order to calculate Cramer’s rule, first, you must have a sound knowledge about determinants.

Determinants are scalar quantities and are functions of square matrices. Determinants are only applicable on square matrices. Determinant allows to understand the inverse and nature of matrices. 

 The formula of the determinant for 2×2 matrix.

a b c d = (a x d) – (b x c)

You can also find the determinant by using an online Determinant Calculator.

The formula of the determinant for 3×3 matrix.

The method of expansion of the 3×3 matrix is given below.

Example

 Find the determinant of the given matrix.

Solution

Step 1: Write the formula of determinant.

a b c d = (a x d) – (b x c)

Step 2: Put the values of a, b, c, and d.

4 6 3 2 = (4 x 2) – (6 x 3)

4 6 3 2 = 8 – 18

4 6 3 2 = -10

Now let us take some examples of Cramer’s rule in order to understand how to calculate the Cramer’s rule. 

Example 1

Solve the linear equation by Cramer’s rule.

5x – y = 3

2x – 3y = 4

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Solution 

Step 1: First write in AX = B form.

5 -1 2 -3 x   y    = 3   4   

Step 2:  Find the determinant of square matrix A.

Step 3: Write the formula of determinant.

a b c d = (a x d) – (b x c)

Step 4: Put the values of a, b, c, and d.

5 -1 2 -3 = (5 x -3) – (2 x -1)

5 -1 2 -3 = -15 – (-2)

5 -1 2 -3 = -15 + 2

5 -1 2 -3 = -13

det = -13

Step 5: Replace the left column of the matrix with the B matrix column or answer column and find the determinant and this matrix is known as the x matrix.

Step 6: Put the values of a, b, c, and d in the determinant formula.

3 -1 4 -3 = (3 x -3) – (4 x -1)

3 -1 4 -3 = -9 – (-4)

3 -1 4 -3 = -9 + 4

3 -1 4 -3 = -5

detx = -5

Step 7: Replace the right column of the matrix with the B matrix column or answer column and find the determinant and this matrix is known as the x matrix.

Step 8: Put the values of a, b, c, and d in the determinant formula.

5 3 2 4 = (5 x 4) – (2 x 3)

5 3 2 4 = 20 – (6)

5 3 2 4 = 20 – 6

5 3 2 4   = 14

dety = 14

Step 9: According to Cramer’s rule we have,

X = detx/det

Y = dety/det

Step 10: Put the values in this formula.

X = -5/-13 = 5/13 = 0.383

Y = 14/-13 = -1.077

You can verify the result by using an online Cramer’s Rule Calculator.

Example 2

Solve the linear equation by Cramer’s rule.

5x + 3y = 4

2x + 6y = 5

Solution 

Step 1: First write in AX = B form.

5 3 2 6 x   y    = 4   5   

Step 2:  Find the determinant of square matrix A.

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Step 3: Write the formula of determinant.

a b c d = (a x d) – (b x c)

Step 4: Put the values of a, b, c, and d.

5 3 2 6 = (5 x 6) – (2 x 3)

5 3 2 6 = 30 – (6)

5 3 2 6 = 30 – 6

5 3 2 6 = 24

det = 24

Step 5: Replace the left column of the matrix with the B matrix column or answer column and find the determinant and this matrix is known as the x matrix.

Step 6: Put the values of a, b, c, and d in the determinant formula.

4 3 5 6 = (4 x 6) – (5 x 3)

4 3 5 6 = 24 – (15)

4 3 5 6   = 24 – 15

4 3 5 6   = 9

detx = 9

Step 7: Replace the right column of the matrix with the B matrix column or answer column and find the determinant and this matrix is known as the x matrix.

Step 8: Put the values of a, b, c, and d in the determinant formula.

5 4 2 5 = (5 x 5) – (2 x 4)

5 4 2 5 = 25 – (8)

5 4 2 5 = 25 – 8

5 4 2 5 = 17

dety = 17

Step 9: According to Cramer’s rule we have,

X = detx/det

Y = dety/det

Step 10: Put the values in this formula.

X = 9/24 = 0.375

Y = 17/24 = 0.708

Summary 

Cramer’s rule is a type of matrix for solving simultaneous linear equations. It is a pretty simple method to solve linear problems. By Cramer’s rule, we can easily find x and y as this method is not complicated. You must have sound knowledge about determinants as in Cramer’s rule determinants are used. Once you grab the knowledge of this topic you will be master it.

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