Use crammers rule to solve the system of linear equations
3x+2y-z = 0
2x-y+z = 1
x-y+2y= -1
Cramer's rule is well explained here:
http://www.purplemath.com/modules/cramers.htm
You will need to calculate the determinants of four 3x3 matrices. I assume you know how to do that. If not, it's high time to learn how.
To solve the system of linear equations using Cramer's Rule, we first need to determine the determinant of the coefficient matrix, denoted as D. Then, we will find the determinants of the matrices obtained by replacing each column of the coefficient matrix with the column of constants, denoted as D_x, D_y, and D_z. Finally, we can find the values of x, y, and z by dividing these determinants by D.
1. Coefficient matrix (A):
| 3 2 -1 |
| 2 -1 1 |
| 1 -1 2 |
2. Calculate the determinant of A (D):
D = 3(-1)(2) + 2(1)(1) + (-1)(2)(-1) - (-1)(-1)(1) - 2(1)(2) - 3(1)(-1)
= -6 + 2 + 2 - 1 - 4 + 3
= -4
3. Create the matrices D_x, D_y, and D_z by replacing the corresponding columns of A with the column of constants:
D_x = | 0 2 -1 |
| 1 -1 1 |
| -1 -1 2 |
D_y = | 3 0 -1 |
| 2 1 1 |
| 1 -1 2 |
D_z = | 3 2 0 |
| 2 -1 1 |
| 1 -1 -1 |
4. Calculate the determinants of D_x, D_y, and D_z:
D_x = 0(-1)(2) + 2(1)(1) + (-1)(-1)(-1) - (-1)(-1)(1) - 2(1)(-1) - 0(1)(-1)
= 0 + 2 + 1 - 1 + 2 + 0
= 4
D_y = 3(1)(2) + 0(1)(1) + (-1)(-1)(2) - (-1)(1)(2) - 0(-1)(1) - 3(-1)(2)
= 6 + 0 + 2 + 2 + 0 + 6
= 16
D_z = 3(-1)(-1) + 2(1)(1) + 0(-1)(-1) - 0(1)(-1) - 2(-1)(1) - 3(1)(1)
= 3 + 2 + 0 + 0 + 2 - 3
= 4
5. Calculate the values of x, y, and z using Cramer's Rule:
x = D_x / D
= 4 / -4
= -1
y = D_y / D
= 16 / -4
= -4
z = D_z / D
= 4 / -4
= -1
Therefore, the solution to the system of linear equations is x = -1, y = -4, z = -1.