Question

Verify A(adj A) = (adj A)A = |A|I.

`[(2,3),(-4,-6)]`

Answer 1

Let, A = `[(2,3),(-4,-6)]`

|A| = −12 + 12

|A| = 0

A₁₁ = (−1)¹⁺¹ M₁₁ = −6

A₁₂ = (−1)¹⁺² M₁₂ = −(−4) = 4

A₂₁ = (−1)²⁺¹ M₂₁ = −3

A₂₂ = (−1)²⁺² M₂₂ = 2

Adj A = `[(-6,-3),(4,2)]`

L.H.S. = A(adj A) = `[(2,3),(-4,-6)] [(-6,-3),(4,2)]`

= `[(2 xx (- 6) + 3 xx 4, 2 xx (-3) + 3 xx 2),(- 4 xx (- 6) + (- 6) xx 4, - 4 xx (- 3) + (- 6) xx 2)]`

= `[(-12 + 12, -6 + 6),(24 - 24, 12 - 12)]`

= `[(0,0),(0,0)]`

R.H.S. = (adj A)A `= [(-6,-3),(4,2)][(2,3),(-4,-6)]`

= `[(-12 + 12,-18 + 18),(8 - 8, 12 - 12)]`

= `[(0,0),(0,0)]`

|A| · I = `0 · [(1,0),(0,1)]`

= `[(0,0),(0,0)]`

Hence, A(adj A) = (adj A)A = |A| · I