Question

Find the inverse of the matrices (if it exists).

`[(2,1,3),(4,-1,0),(-7,2,1)]`

Answer 1

Let A = `[(2,1,3),(4,-1,0),(-7,2,1)]`

Then |A| = `|(2,1,3),(4,-1,0),(-7,2,1)|`

= 2(−1 − 0) − 1(4 − 0) + 3(8 − 7)

= −2 − 4 + 3

= −3 ≠ 0

So, A is a non-singular matrix and therefore, it is invertible. Let C_ij be cofactor of a_ij in A. Then the cofactors of elements of A are given by,

C₁₁ = `(-1)^(1+1) |(-1,0), (2,1)|`

= 1 × (−1 − 0)

= −1

C₁₂ = `(-1)^(1+2) |(4,0), (-7,1)|`

= (−1) × (4 + 0)

= −1 × 4

= −4

C₁₃ = `(-1)^(1+3)|(4,-1),(-7,2)|`

= 1 × (8 − 7)

= 1 × 1

= 1

C₂₁ = `(-1)^(2+1) |(1,3), (2,1)|`

= (−1) × (1 − 6)

= (−1) × (−5)

= 5

C₂₂ = `(-1)^(2+2) |(2,3), (-7,1)|`

= 1 × (2 + 21)

= 1 × 23

= 23

C₂₃ = `(-1)^(2+3) |(2,1), (-7,2)|`

= (−1) × (4 + 7)

= (−1) × 11

= −11

C₃₁ = `(-1)^(3+1) |(1,3), (-1,0)|`

= 1 × (0 + 3)

= 1 × 3

= 3

C₃₂ = `(-1)^(3+2) |(2,3), (4,0)|`

= (−1) × (0 − 12)

= (−1) × (−12)

= 12

C₃₃ = `(-1)^(3+3)|(2,1), (4,-1)|`

= 1 × (−2 − 4)

= 1 × (−6)

= −6

∴ adj A = `[(-1,-4,1),(5,23,-11),(3,12,-6)] = [(-1,5,3),(-4,23,12),(1,-11,-6)]`

A⁻¹ = `1/|A|` adj A

= `1/-3 [(-1,5,3),(-4,23,12),(1,-11,-6)]`