Find the inverse M^{-1} of M = [ [2, 1], [5, 3] ].

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Multiple Choice

Find the inverse M^{-1} of M = [ [2, 1], [5, 3] ].

Explanation:
The idea is to use the inverse formula for a 2x2 matrix: if A = [ [a, b], [c, d] ], then A^{-1} = (1/(ad − bc)) [ [d, −b], [−c, a] ], provided the determinant ad − bc is not zero. Here a = 2, b = 1, c = 5, d = 3. The determinant is ad − bc = 2·3 − 1·5 = 6 − 5 = 1. The adjugate is [ [d, −b], [−c, a] ] = [ [3, −1], [−5, 2] ]. Since the determinant is 1, the inverse is the adjugate itself: [ [3, −1], [−5, 2] ]. A quick check confirms the result: [ [2, 1], [5, 3] ] times [ [3, −1], [−5, 2] ] equals the identity matrix.

The idea is to use the inverse formula for a 2x2 matrix: if A = [ [a, b], [c, d] ], then A^{-1} = (1/(ad − bc)) [ [d, −b], [−c, a] ], provided the determinant ad − bc is not zero.

Here a = 2, b = 1, c = 5, d = 3. The determinant is ad − bc = 2·3 − 1·5 = 6 − 5 = 1. The adjugate is [ [d, −b], [−c, a] ] = [ [3, −1], [−5, 2] ]. Since the determinant is 1, the inverse is the adjugate itself: [ [3, −1], [−5, 2] ].

A quick check confirms the result: [ [2, 1], [5, 3] ] times [ [3, −1], [−5, 2] ] equals the identity matrix.

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