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Question
Find the inverse of the following matrix:
`[(-3,-5,4),(-2,3,-1),(1,-4,-6)]`
Solution
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
[Aij] = `[(-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 = [Aij]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)]`
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