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Question
Find the inverse of the following matrices by the adjoint method `[(2, -2),(4, 5)]`.
Solution
Let A = `[(2, -2),(4, 5)]`
∴ |A| = `[(2, -2),(4, 5)]` = 10 + 8 = 18 ≠ 0
∴ A–1 exists.
A11 = (– 1)1+1 M11 = (1)(5) = 5
A12 = (– 1)1+2 M12 = (– 1)(4) = – 4
A21 = (– 1)2+1 M21 = (– 1)(– 2) = 2
A22 = (– 1)2+2 M22 = (1)(2) = 2
∴ The matrix of the co-factors is
[Aij]2x2 = `[("A"_11, "A"_12),("A"_21, "A"_22)] = [(5, -4),(2, 2)]`
Now adj A = `["A"_"ij"]_(2xx2)^"T" = [(5, 2),(-4, 2)]`
∴ A–1 = `(1)/|"A"|("adj A")`
= `(1)/(18)[(5, 2),(-4, 2)]`.
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