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Question
Find the inverse of the following matrices by the adjoint method `[(3, -1),(2, -1)]`.
Solution
Let A = `[(3, -1),(2, -1)]`
∴ |A| = `|(3, - 1),(2, -1)|` = – 3 + 2 = – 1 ≠ 0
∴ A–1 exists.
A11 = (– 1)1+1 M11 = (1)(–1) = – 1
A12 = (– 1)1+2 M12 = (– 1)(2) = – 2
A21 = (– 1)2+1 M21 = (– 1)(– 1) = 1
A22 = (– 1)2+2 M22 = (1)(3) = 3
∴ The matrix of the co-factors is
[Aij]2x2 = `[("A"_11, "A"_12),("A"_21, "A"_22)] = [(-1, -2),(1, 3)]`
Now adj A = `["A"_"ij"]_(2xx2)^"T" = [(-1, 1),(-2, 3)]`
∴ A–1 = `(1)/|"A"|("adj A")`
= `(1)/(-1)[(-1, 1),(-2, 3)]`
= `[(1, -1),(2, -3)]`.
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