Question

Let A be a matrix in O3 whose determinant is -1. Prove that -1
is an eigenvalue of A.

Answer 1

To prove that -1 is an eigenvalue of a matrix A in O3 with determinant -1, we can show that the determinant of A + I is 0, where I is the identity matrix.

Step-by-step explanation:

Let A be a matrix in O3 with determinant -1. To prove that -1 is an eigenvalue of A, we need to show that there exists a non-zero vector v such that Av = -v.

Since A is in O3, its determinant is either 1 or -1. Since we are given that the determinant is -1, it implies that the determinant of A + I is 0, where I is the identity matrix.

Therefore, there exists a non-zero vector v such that (A + I)v = 0. Expanding this equation, we get Av + v = 0, which means Av = -v. Thus, -1 is an eigenvalue of A.