A is a real symmetric matrix of third order, a ^ 2 + 2A = 0, R (a) = 2. Find all eigenvalues of a and the value of determinant | a ^ 2 + 3E | Why R (a) = 2, then - 2 is a double root?

A is a real symmetric matrix of third order, a ^ 2 + 2A = 0, R (a) = 2. Find all eigenvalues of a and the value of determinant | a ^ 2 + 3E | Why R (a) = 2, then - 2 is a double root?

This is because "the rank of a diagonalizable matrix is equal to the number of its nonzero eigenvalues"
A is a real symmetric matrix, a (a + 2e) = 0, so the eigenvalues of a can only be 0, - 2
The eigenvalues of a are 0, - 2, - 2 from R (a) = 2
So the eigenvalues of a ^ 2 + 3E are (λ ^ 2 + 3): 3,7,7
So | a ^ 2 + 3E | = 3 * 7 * 7 = 147