# Cramer's Rule, solving system of linear equations

I must solve the following linear equations using matrix methods.

x+y-z=-8

3x-y+z=-4

-x+2y+2z=21

I am trying to understand the method of solving for variables of linear equation by forming them into a matrix and solving for the variables. Please help.

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## SOLUTION This solution is **FREE** courtesy of BrainMass!

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There are three methods of solving systems of linear equations using matrices.

1. Row elimination method.

2. Cramer's Rule

3. Inverse matrix method.

Here the question is x+y-z=-8

3x-y+z=-4

-x+2y+2z=21

Now we have to find the value of x, y and z using any one of the matrices method.

Here we are going to use the second method called Cramer's rule.

Step:1

Write the coefficient matrix of the system (call this matrix A); if it is square matrix (square matrix is nothing but equal number of rows and columns) , you may continue, otherwise Cramer's rule is not applicable here.

Here we can continue with cramer's rule because the given matrix is square matrix.

The co efficient matrix of this system is

1 1 -1

A= 3 -1 1

-1 2 2

That is we have written the co-efficient of x,y, and z in matrix form.

Now we have to find the determinant of this matrix. I hope that you will familiar with finding the determinant of the matrix.

det(A)= 1(-2-2)-1(6+1)-1(6-1)

= 1(-4)-1(7)-1(5)

= -4-7-5

det(A)= -16 So, the det(A) is -16

We can write determinant as l A l = -16. Here determinant not equal to zero so we can use the cramer's rule. Suppose if l A l= 0 we cannot use the cramer's rule:)

Step : 2

Here the first variable of the system is x. Then write the matrix A as follows:

(x)

Substitute the column of numbers to the right if the equal signs instead of the first column of A.

-8 1 -1

That is A = -4 -1 1

(x) 21 2 2

Here compare A and A only the x terms interchanged .

(x)

Now we have to find the determinant of A , as we found det(A).

(x)

A = -8(-2-2)-1(-8-21)-1(-8+21)

(x)

= -8(-4)-1(-29)-1(13)

= 32+29-13

A = 48 , This can be written as l A l

(x) (x)

To find the value of x, We have to divide l A l by lAl.

(x)

That is x = l A l

(x)

______

l A l

So, x = 48 / -16 = -3

That is , the value of x =-3.

Similarly we have to find the y and z.

Step :4

Now find l A (y) l from A(y) = 1 -8 -1

3 -4 1

-1 21 2

l A(y) l= 1(-8-21) +8 (6+1)-1(63-4)

= -32

So, the value of y is lA(y)l divide by lAl , That is y= -32/-16 =2.

y= 2

Step: 5

Now finally we have to find the value of z.

Find lA(z)l from A(z) = 1 1 -8

3 -1 -4

-1 2 21

lA(z)l= 1(-21+8)-1(63-4) -8(6-1)

lA(z)l = -112.

So, the value of z is lA(z)l divide by lAl.

That is, z= -112/-16 = 7

Hence, the value of x = -3, y= 2 and z= 7.

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