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Question
Find the inverse of the matrix `[(1 2 3),(1 1 5),(2 4 7)]` by adjoint method
Solution
Let A = `[(1 2 3),(1 1 5),(2 4 7)]`
| A11 = (-1)1+1 | M11 = (-1)2 (7 - 20) = -13 |
|
A12 = (-1 )1+2 |
M12 = (-1)3 (7 - 10) = 3 |
| A13 = (-1 )1+3 |
M13 = (-1)4 (4 - 2) = 2 |
| A21 = (-1 )2+1 | M21 = (-1)3 (14 - 12) = -2 |
| A22 = (-1 )2+2 | M22 = (-1)4 (7 - 6) = 1 |
| A23 = (-1 )2+3 | M23 = (-1)5 (4 - 4) = 0 |
| A31 = (-1 ) 3+1 | M31 = (-1)4 (10 - 3) = 7 |
| A32 = (-1 )3+2 | M32 = (-1)5 (5 - 3) = -2 |
| A33 = (-1 )3+3 | M33 = (-1)6 (1 - 2) = -1 |
∴ Matrix of cofactor = `[(-13,3,2),(-2,1,0),(7,-2, -1)]`
Adj A = Transpose of the cofactor matrix [cij]
i.e. Adj A = [cij]' = `[(-13,-2,7),(3,1 ,-2),(2,0,-1)]`
Now , determinant of A is |A| = `[(1,2,3),(1 ,1, 5),(2 ,4 ,7)]`
= 1(7 - 20) - 2(7 - 10) + 3(4 - 2)
= 1(-13) - 2(-3) + 3(2)
= -13 + 6 + 6 = -1
∴ |A| = -1 ≠ 0
Now inverse of A is A-1 = `1/|"A"|` × Adj A
= `1/-1 [(-13,-2,7),(3,1,-2),(2,0,-1)]`
∴ A-1 = `[(13,2,-7),(-3,-1,2),(-2,0,1)]`
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