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If A = [[5a, -b], [3, 2]] and A adj A = A A^T, then 5a + b is equal to:

1. 4
2. 13
3. -1
4. 5

Correct answer: 5

Solution

A adj A = det(A) I = (10a+3b)I. A A^T = [[25a^2+b^2, 15a-2b],[15a-2b,13]]. Off-diagonal: 15a-2b=0. Diagonal: 13=10a+3b. Solving gives a=2/5, b=3, so 5a+b = 2+3 = 5.

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