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प्रश्न
If A = `|(3,-1,1),(-15,6,-5),(5,-2,2)|` then, find the Inverse of A.
उत्तर
A = `|(3,-1,1),(-15,6,-5),(5,-2,2)|`
`= 3|(6,-5),(-2,2)| + 1|(-15,-5),(5,2)| + 1|(-15,6),(5,-2)|`
= 3(12 - 10) + 1(- 30 + 25) + 1(30 - 30)
= 6 - 5 = 1 ≠ 0
∴ A-1 exists.
adj A = `[(+|(6,-5),(-2,2)|, -|(-15,-5),(5,2)|, +|(-15,6),(5,-2)|),(-|(-1,1),(-2,2)|, +|(3,1),(5,2)|, -|(3,-1),(5,-2)|),
(+|(-1,1),(6,-5)|, -|(3,1),(-15,-5)|, +|(3,-1),(-15,6)|)]^"T"`
`= [(+(12-10),-(-30 + 25),+(30-30)),(-(-2+2),+(6-5),-(-6+5)),(+(5-6),-(-15+15),+(18-15))]^"T"`
`= [(2,5,0),(0,1,1),(-1,0,3)]^"T"`
`= [(2,0,-1),(5,1,0),(0,1,3)]`
Now, `"A"^-1 = 1/|"A"|`adj A
`= 1/1 [(2,0,-1),(5,1,0),(0,1,3)] = [(2,0,-1),(5,1,0),(0,1,3)]`
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