Question

If A = `[(2, -3, 5),(3, 2, -4),(1, 1, -2)]`, then find A^–1. Using A^–1, solve the following system of equations:

2x – 3y + 5z = 11

3x + 2y – 4z = –5

x + y – 2z = –3

Answer 1

Given, A = `[(2, -3, 5),(3, 2, -4),(1, 1, -2)]`

|A| = `|(2, -3, 5),(3, 2, -4),(1, 1, -2)|`

= 2(–4 + 4) + 3(–6 + 4) + 5(3 – 2)

= 3(–2) + 5(1)

= –1 ≠ 0

∴ A^–1 exists

Suppose A_ij is the cofactor of element a_ij of A

Now, A₁₁ = 0, A₁₂ = –(–2) = 2, A₁₃ = 1,

A₂₁ = –1, A₂₂ = –9, A₂₃ = –5,

A₃₁ = 2, A₃₂ = 23,

A₃₃ = 13

∴ Adj A = `[(0, -1, 2),(2, -9, 23),(1, -5, 13)]`

`A^-1 = 1/|A| xx Adj A`

= `[(0, 1, -2),(-2, 9, -23),(-1, 5, -13)]`

Also, we need to solve the following system of equations

2x – 3y + 5z = 11

3x + 2y – 4z = –5

x + y – 2z = –3

Here, A = `[(2, -3, 5),(3, 2, -4),(1, 1, -2)]` B = `[(11),(-5),(-3)]`, X = `[(x),(y),(z)]`

∴ X = A^–1B

`[(x),(y),(z)] = [(0, 1, -2),(-2, -, -23),(-1, 5, -13)] [(11),(-5),(-3)]`

⇒ `[(x),(y),(z)] = [(0 xx 11 + 1 xx (-5) + (-2) xx (-3)),((-2) xx 11 + 9 xx (-5) + (-23) xx (-3)),((-1) xx 11 + 5 xx (-5) + (-13) xx (-3))] = [(1),(2),(3)]`

From equality of matrices,

x = 1, y = 2, z = 3