Question 86

Let A be a $$2 \times 2$$ real matrix and I be the identity matrix of order 2. If the roots of the equation $$|A - xI| = 0$$ be -1 and 3, then the sum of the diagonal elements of the matrix $$A^2$$ is _____.

Correct Answer: 10

A is 2×2 with eigenvalues -1 and 3.

Trace of A = -1 + 3 = 2, det(A) = (-1)(3) = -3.

By Cayley-Hamilton: $$A^2 - 2A - 3I = 0$$, so $$A^2 = 2A + 3I$$.

Trace of $$A^2$$ = 2·trace(A) + 3·trace(I) = 2(2) + 3(2) = 4 + 6 = 10.

The answer is $$\boxed{10}$$.

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