Question

Solve the following system of equations by matrix method:
 x + y + z = 6
x + 2z = 7
3x + y + z = 12

Answer 1

 x + y + z = 6
x + 2z = 7
3x + y + z = 12

AX = B

`A = [(1,1,1),(1,0,2),(3,1,1)], X = [(x),(y),(z)], B = [(6),(7),(12)]`

X = A⁻¹B

`A^-1 = 1/|A| . adj(A)`

`A = [(1,1,1),(1,0,2),(3,1,1)]`

Use cofactor expansion along the first row:

`|A| = 1 . |(0,2),(1,1)|-1. |(1,2),(3,1)| +1.|(1,0),(3,1)|`

Now calculate each 2 × 2 determinant:

`|(0,2),(1,1)| = 0(1)-2(1) = -2`

`|(1,2),(3,1)| = 1(1) -2(3) = 1-6=-5`

`|(1,0),(3,1)| = 1(1) - 0(3) = 1`

∣A∣ = 1(−2)−1 (−5) + 1(1) = −2 + 5 + 1 = 4

We already computed these minors earlier, but here’s the full cofactor matrix:

`Cof(A) = [(-2,5,1),(0,0,-2),(2,-1,-1)]`

adj(A) = Cof(A)^T = `[(-2,0,2),(5,0,-1),(1,-2,-1)]`

`A^-1 = 1/4 . [(-2,0,2),(5,0,-1),(1,-2,-1)]`

X = A⁻¹B = `1/4 [(-2,0,2),(5,0,-1),(1,-2,-1)] . [(6),(7),(12)]`

(−2) (6) + (0) (7) + (2) (12) = −12 + 0 + 24 = 12

(5) (6) + (0) (7) + (−1) (12) = 30 + 0 −12 = 18

(1) (6) + (−2) (7) + (−1) (12) = 6 − 14 − 12 = −20

`X = 1/4 . [(12),(18),(-20)] = [(3),(4.5),(-5)]`

x = 2, y = 1, z = 3