Using matrices, solve the following system of linear equations:

x - y + 2z = 7

3x + 4y - 5z = - 5

2x - y + 3z = 12

OR

Using elementary operations, find the inverse of the following matrix:

Given; x - y + 2z = 7

3x + 4y - 5z = - 5

2x - y + 3z = 12

∴ AX = B

⇒ X = A ^{- 1}B

|A| = 1(12 − 5) + 1(9 + 10) + 2(−3 − 8)

= 7 + 19 − 22

= 4 ≠ 0

∴ A ^{- 1} exists

Cofactors of A are;

A₁₁ = 7 A₁₂ = −19 A₁₃ = −11

A₂₁ = 1 A₂₂ = −1 A₂₃ = −1

A₃₁ = −3 A₃₂ = 11 A₃₃ = 7

∴ x = 2, y = 1, z = 3 is the Required solution.

OR

Given;

A = IA

R₁→ 2R₁ + R₃

R₂→ R₂ − R₁

R₂→ R₂ − R₁

R₁ → R₁ + 3R₂ and R₃ → R₃ − 3R₁

R₂ → −R₂

R₃ → R₃ + 8R₂

R₁ → R₁ + 1/2 R₃

R₂ → R₂ − R₃ and R₃ → 1/2 R₃