Calculus eigenvalues

Question

The matrix A = [−3, k] has two distinct real eigenvalues if and only if k < 

[−2−7]

Answer 1

A is the 2x2 matrix with first row [-3, k] and second row [-2, -7].

The characteristic polynomial of A is det(A - λI) = 0, where I is the 2x2 identity matrix.

So, the characteristic polynomial in this case is λ² + 10λ + (21+2k) = 0

The matrix, A, has 2 distinct real eigenvalues only when the discriminant of the characteristic polynomial is positive. [Recall that the discriminant of Ax² + Bx + C = 0 is B² - 4AC].

So, we must have (10)² - 4(1)(21+2k) > 0.

100 - 84 - 8k > 0

16 - 8k > 0

k < 2