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Question
Find the inverse of the following matrix by the adjoint method.
`[(2,-2),(4,3)]`
Solution
Let A = `[[2,-2],[4,3]]`
|A|= `|[2,-2],[4,3]|` = 6 + 8 = 14 `≠` 0
∴ A−1 exist
First we have to find the co-factor matrix
= [Aij]2×2′, where Aij = (−1)i+jMij
Now, A11 = (− 1)1+1M11 = 3
A12 = (− 1)1+2M12 = − 4
A21 = (− 1)2+1M21 = (− 2) = 2
A22 = (− 1)2+2M22 = 2
Hence, the co-factor matrix
= `[("A"_11,"A"_12),("A"_21,"A"_22)]` = `[(3,-4),(2,2)]`
∴ adj A `=[(3,2),(-4,2)]`
∴ A−1 = `1/|"A"|` (adj A) = `1/14[(3,2),(-4,2)]`
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