Question

If A^-1 = `[(1,0,3),(2,1,-1),(1,-1,1)]`  then, find A.

Answer 1

Given A^-1 = `[(1,0,3),(2,1,-1),(1,-1,1)]`

We know that (A^-1)^-1 = A

So we have to find inverse of A^-1

`|"A"^-1| = |(1,0,3),(2,1,-1),(1,-1,1)|`

= 1(1 - 1)- 0(2 + 1) + 3(-2 - 1)

= 1(0) - 0(03 + 3(-3)

= 0 - 0 - 9 = - 9 ≠ 0

`["A"_"ij"^-1] = [(0,-3,-3),
(-|(0,3),(-1,1)|,|(1,3),(1,1)|,-|(1,0),(1,-1)|),
(|(0,3),(1,-1)|,-|(1,3),(2,-1)|,|(1,0),(2,1)|)]`

= `[(0,-3,-3),(-(0+3),1-3,-(-1-0)),(0-3,-(-1-6),(1-0))]`

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

adj A^-1 = `["A"_"ij"^-1]^"T" = [(0,-3,-3),(-3,-2,7),(-3,1,1)]`

∴ (A^-1)^-1 = `1/|"A"^-1|` (adj A^-1)

`= 1/(-9)[(0,-3,-3),(-3,-2,7),(-3,1,1)]`

i.e., A = `1/9[(0,3,3),(3,2,-7),(3,-1,-1)]`