Question

If A = `[(1, 2),(3, 4)]` verify that A (adj A) = (adj A) A = |A| I

Answer 1

For A = `[(1, 2),(3, 4)]`

A₁₁ = (–1)^{1 + 1} (4) = 4

A₁₂ = (–1)^{1 + 2} (3) = –3

A₂₁ = (–1)^{2 + 1} (2) = –2

A₂₂ = (–1)^{2 + 2} (1) = 1

adj A = `[(A_11, A_21),(A_12, A_22)]`

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

∴ A(adj A) = `[(1, 2),(3, 4)][(4, -2),(-3, 1)]`

=  `[(4 - 6, -2 +2),(12 -12, -6 + 4)]`

= `[(-2, 0),(0, -2)]`  ...(i)

(adj A) . A = `[(4, -2),(-3, 1)][(1, 2),(3, 4)]`

= `[(4 - 6, 8 - 8),(-3 + 3, -6 + 4)]`

= `[(-2, 0),(0, -2)]`  ...(ii)

and |A| I = `[(1, 2),(3, 4)][(1, 0),(0, 1)]` = 4 -6 = -2

= `(-2) [(1, 0),(0, 1)]`

= `[(-2, 0),(0, -2)]`  ...(iii)

From (i), (ii) and (iii) we get,

A(adj A) = (adj A) A = |A| I