Question

If A = `|(1,1,1),(3,4,7),(1,-1,1)|` verify that A(adj A) = (adj A)(A) = |A|I₃.

Answer 1

Given A = `|(1,1,1),(3,4,7),(1,-1,1)|`

|A| = `1|(4,7),(-1,1)| -1|(3,7),(1,1)| +1|(3,4),(1,-1)|`

= (4 + 7) - 1(3 - 7) + 1(-3 - 4)

= 11 + 4 - 7 = 8

adj A = `[(+ |(4,7),(-1,1)|,-|(3,7),(1,1)|,+|(3,4),(1,-1)|),(-|(1,1),(-1,1)|,+|(1,1),(1,1)|,-|(1,1),(1,-1)|),(+|(1,1),(4,7)|,-|(1,1),(3,7)|,+|(1,1),(3,4)|)]^"T"`

`= [((4+7),-(3-7),+(-3-4)),(-(1+1),+(1-1),-(-1-1)),(+(7-4), -(7-3), +(4-3))]^"T"`

`= [(11,4,-7),(-2,0,2),(3,-4,1)]^"T"`

`= [(11,-2,3),(4,0,-4),(-7,2,1)]`

Now

A(adj A) = `[(1,1,1),(3,4,7),(1,-1,1)][(11,-2,3),(4,0,-4),(-7,2,1)]`

`= [(11+4-7,-2+0+2,3-4+1),(33+16-49,-6+0+14,9-16+7),(11-4-7,-2+0+2,3+4+1)]`

`= [(8,0,0),(0,8,0),(0,0,8)]`    ....(1)

(adj A)A = `[(11,-2,3),(4,0,-4),(-7,2,1)] [(1,1,1),(3,4,7),(1,-1,1)]`

`= [(11-6+3,11-8-3,11-14+3),(4+0-4,4+0+4,4+0-4),(-7+6+1,-7+8-1,-7+14+1)]`

`= [(8,0,0),(0,8,0),(0,0,8)]`    ....(2)

|A|.I₃ = `8[(1,0,0),(0,1,0),(0,0,1)] = [(8,0,0),(0,8,0),(0,0,8)]`   .....(3)

From (1), (2) and (3)

A(adj A) = (adj A)(A) = |A| I₃.