Using matrices, solve the following system of equations: x + y - z = 3; 2x + 3y + z = 10; 3x - y - z = 1

The given system of equation can be expressed can be represented in matrix form as AX = B, where
A = ∣∣∣∣1 1 -1
2 3 1
3 -1 -1∣∣∣∣, X = ∣∣∣∣x
y
z∣∣∣∣, B = ∣∣∣∣3
10
1∣∣∣∣
Now |A| = ∣∣∣∣1 1 -1
2 3 1
3 -1 -1∣∣∣∣ = 1(3(-1) + 1) - 1(2(-1) - 3) + (-1)(-2 - 9) = -2 + 5 + 11 = 14 ≠ 0
The solution is given by X = A⁻¹B where A⁻¹ = (1/|A|) adj(A)
To find adj(A), we find the cofactor matrix of A and then transpose it.
C₁₁ = (3)(-1) + 1 = -2
C₁₂ = -(2(-1) - 3) = 5
C₁₃ = (2)(-1) - (3)(3) = -11
C₂₁ = -(1(-1) + 1) = 0
C₂₂ = (-1)(-1) - (-1)(3) = 4
C₂₃ = -(1(-1) - 1(3)) = 4
C₃₁ = (1(1) - (-1)(3)) = 4
C₃₂ = -(1(1) - (-1)(2)) = -3
C₃₃ = (1(3) - (1)(2)) = 1
Cofactor matrix = ∣∣∣∣-2 5 -11
0 4 4
4 -3 1∣∣∣∣
adj(A) = ∣∣∣∣-2 0 4
5 4 -3
-11 4 1∣∣∣∣
A⁻¹ = (1/14) ∣∣∣∣-2 0 4
5 4 -3
-11 4 1∣∣∣∣
X = A⁻¹B = (1/14) ∣∣∣∣-2 0 4
5 4 -3
-11 4 1∣∣∣∣ ∣∣∣∣3
10
1∣∣∣∣ = (1/14) ∣∣∣∣-6 + 0 + 4
15 + 40 - 3
-33 + 40 + 1∣∣∣∣ = (1/14) ∣∣∣∣-2
52
8∣∣∣∣ = ∣∣∣∣-1/7
26/7
4/7∣∣∣∣
Therefore, x = -1/7, y = 26/7, z = 4/7