Let A = [[1,2],[3,4]]. Which of the following is A^{-1}?

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Multiple Choice

Let A = [[1,2],[3,4]]. Which of the following is A^{-1}?

To find the inverse of a 2x2 matrix, use the determinant and the adjugate. For A = [[1, 2], [3, 4]], the determinant is det(A) = 1·4 − 2·3 = −2, which is nonzero, so an inverse exists. The inverse is (1/det) times [[d, −b], [−c, a]] with a = 1, b = 2, c = 3, d = 4. That gives (1/−2)·[[4, −2], [−3, 1]] = [[−2, 1], [3/2, −1/2]]. This matches the correct result, and you can check quickly by multiplying A by this matrix to see you get the identity: A·A^{-1} = [[1,2],[3,4]] · [[−2, 1], [3/2, −1/2]] = I.

To find the inverse of a 2x2 matrix, use the determinant and the adjugate. For A = [[1, 2], [3, 4]], the determinant is det(A) = 1·4 − 2·3 = −2, which is nonzero, so an inverse exists. The inverse is (1/det) times [[d, −b], [−c, a]] with a = 1, b = 2, c = 3, d = 4. That gives (1/−2)·[[4, −2], [−3, 1]] = [[−2, 1], [3/2, −1/2]].

This matches the correct result, and you can check quickly by multiplying A by this matrix to see you get the identity:

A·A^{-1} = [[1,2],[3,4]] · [[−2, 1], [3/2, −1/2]] = I.

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