# Find the inverse of the following matrices by the adjoint method [2-245]. - Mathematics and Statistics

Sum

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)].

Concept: Inverse of Matrix
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# Find the inverse of the following matrices by the adjoint method [2-245]. - Mathematics and Statistics

Sum

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)].

Concept: Inverse of Matrix
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The text passage contains a mathematical question asking for the inverse of a given matrix using the adjoint method, and it provides a detailed solution to this question.

TYPE: SINGLE_QUESTION_WITH_ANSWER