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Question
Find the inverse of the following matrix by the adjoint method.
`[(-1,5),(-3,2)]`
Solution
Let A = `[(-1,5),(-3,2)]`
∴ |A| = `|(-1,5),(-3,2)|` = − 2 + 15 = 13 `≠` 0
∴ A−1 exists.
First we have to find the co-factor matrix
= [Aij]2×2′ where Aij = (− 1)i+jMij
Now, A11 = (− 1)1+1M11 = 2
A12 = (− 1)1+2M12 = − (− 3) = 3
A21 = (− 1)2+1M21 = − 5
A22 = (− 1)2+2M22 = − 1
Hence, the co-factor matrix
= `[("A"_11,"A"_12),("A"_21,"A"_22)]` = `[(2,3),(-5,-1)]`
∴ adj A = `[(2,-5),(3,-1)]`
∴ A−1 = `1/|"A"|` (adj A) = `1/13[(2,-5),(3,-1)]`
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