Question

Suppose b is a real 2x2 matrix with two complex eigenvalues. What can be said about the determinant of b?

a) It must be a real number.
b) It must be an imaginary number.
c) It can be either real or imaginary.
d) It cannot be determined.

Answer 1

The determinant of a real 2x2 matrix with two complex eigenvalues must be a real number because the eigenvalues come in complex conjugate pairs and their product is a sum of squares of real numbers.

Step-by-step explanation:

When dealing with a real 2x2 matrix b that has two complex eigenvalues, one can say that the determinant of b must be a real number. This can be shown using the fact that the complex eigenvalues of a matrix come in complex conjugate pairs.

If the two complex eigenvalues are a + ib and a - ib, where a and b are real numbers, the determinant of matrix b is the product of its eigenvalues, which would be (a + ib) multiplied by (a - ib), resulting in a2 + b2. Since both aa and b are real numbers, the determinant is also a real number.

Therefore, the correct answer is (a) It must be a real number.

When a real 2x2 matrix has two complex eigenvalues, the determinant of the matrix can be either real or imaginary.

Let's say the eigenvalues of matrix b are a+bi and a-bi, where a and b are real numbers.

The determinant of matrix b is given by (a+bi)*(a-bi) = a^2 - (bi)^2 = a^2 + b^2.

Since a^2 and b^2 are both real numbers, the determinant of matrix b can either be a real number or 0 (if a=b=0).