Question

Find the inverse of the following matrix:

`[(-3,-5,4),(-2,3,-1),(1,-4,-6)]`

Answer 1

Let A = `[(-3,-5,4),(-2,3,-1),(1,-4,-6)]`

|A| = `|(-3,-5,4),(-2,3,-1),(1,-4,-6)|`

`= -3|(3,-1),(-4,-6)| - (-5)|(-2,-1),(1,-6)| + 4|(-2,3),(1,-4)|`

= -3[-18 - 4] - (-5)[12 + 1] + 4[8 - 3]

= - 3[- 22]- (- 5)[13] + 4 [5]

= 66 + 65 20 = 151

[A_ij] = `[(-22,-13,5),(-|(-5,4),(-4,-6)|,|(-3,4),(1,-6)|,-|(-3,-5),(1,-4)|),(|(-5,4),(3,-1)|,-|(-3,4),(-2,-1)|,|(-3,-5),(-2,3)|)]`

`= [(-22,-13,5),(-(30 + 16),(18-4),-(12+5)),(5-12,-(3+8),(-9-10))]`

`= [(-22,-13,5),(-46,14,-17),(-7,-11,-19)]`

adj A = [A_ij]^T = `[(-22,-46,-7),(-13,14,-11),(5,-17,-19)]`

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

`= 1/151[(-22,-46,-7),(-13,14,-11),(5,-17,-19)]`