Question

If A = `[(1,3,3),(1,4,3),(1,3,4)]` then verify that A(adj A) = |A| I and also find A^-1.

Answer 1

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

|A| = `|(1,3,3),(1,4,3),(1,3,4)|`

`= 1|(4,3),(3,4)| - 3|(1,3),(1,4)| + 3|(1,4),(1,3)|`

= 1[16 – 9] – 3[4 – 3] + 3[3 – 4]

= 1(7) – 3(1) + 3(-1)

= 7 – 3 – 3

= 1

Cofactor [A_ij] = `[(7,-1,-1),(-|(3,3),(3,4)|,|(1,3),(1,4)|,-|(1,3),(1,3)|),(|(3,3),(4,3)|,-|(1,3),(1,3)|,|(1,3),(1,4)|)]`

`= [(7,-1,-1),(-(12-9),(4-3),0),((9-12),0,(4-3))]`

`= [(7,-1,-1),(-3,1,0),(-3,0,1)]`

adj A = [A_ij]^T = `[(7,-3,-3),(-1,1,0),(-1,0,1)]`

Now A(adj A) = `[(1,3,3),(1,4,3),(1,3,4)][(7,-3,-3),(-1,1,0),(-1,0,1)]`

`= [(7-3-3,-3+3+0,-3+0+3),(7-4-3,-3+4,-3+0+3),(7-3-4,-3+3+0,-3+0+4)]`

`= [(1,0,0),(0,1,0),(0,0,1)]`   ....(1)

|A| I = `1[(1,0,0),(0,1,0),(0,0,1)] = [(1,0,0),(0,1,0),(0,0,1)]`   .....(2)

`"A"^-1 = 1/|"A"|` adj A

`= 1/1 [(7,-3,-3),(-1,1,0),(-1,0,1)] = [(7,-3,-3),(-1,1,0),(-1,0,1)]`

From (1) and (2),

A(Adj A) = |A| I