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Question
Find the inverse of the following matrix by the adjoint method.
`[(1, 0, 0),(3, 3, 0),(5, 2, -1)]`
Solution
Let A = `[(1, 0, 0),(3, 3, 0),(5, 2, -1)]`
∴ |A| = `|(1, 0, 0),(3, 3, 0),(5, 2, -1)|`
= 1(−3 − 0) − 0 + 0
= −3 ≠ 0
∴ A−1 exist
First we have to find the co-factor matrix
= [Aij]3×3′ where Aij = (− 1)i+jMij
Now, A11 = (−1)1+1M11 = 1`|(3, 0),(2, -1)|` = 1(−3 − 0) = −3
A12 = (−1)1+2M12 = − 1`|(3, 0),(5, -1)|` = −1(−3 − 0) = 3
A13 = (−1)1+3M13 = 1`|(3, 3),(5, 2)|` = 1(6 − 15) = −9
A21 = (−1)2+1M21 = −1`|(0, 0),(2, -1)|` = −1(0 − 0) = 0
A22 = (−1)2+2M22 = 1`|(1, 0),(5, -1)|` = 1(−1 − 0) = −1
A23 = (−1)2+3M23 = −1`|(1, 0),(5, 2)|` = −1(2 − 0) = −2
A31 =(−1)3+1M31 = 1`|(0, 0),(3, 0)|` = 1(0 − 0) = 0
A32 = (−1)3+2M32 = −1`|(1, 0),(3, 0)|` = −1(0 − 0) = 0
A33 = (−1)3+3M33 = 1`|(1, 0),(3, 3)|` = 1(3 − 0) = 3
∴ The co-factor matrix
= `[(A_11, A_12, A_13),(A_21, A_22, A_23),(A_31, A_32, A_33)]` = `[(-3, 3, -9),(0, -1, -2),(0, 0, 3)]`
∴ adj A = `[(-3, 0, 0),(3, -1, 0),(-9, -2, 3)]`
∴ A−1 = `1/|A|` (adj A)
= `-1/3[(-3, 0, 0),(3, -1, 0),(-9, -2, 3)]`
∴ A−1 = `1/3[(3, 0, 0),(-3, 1, 0),(9, 2, -3)]`
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