6 Note that an eigenvector cannot be zero but an eigenvalue

6. Note that an eigenvector cannot be zero, but an eigenvalue can be 0. Suppose that 0 is an eigenvalue of A. What does it say about A? (Hint: One of the most important properties of a matrix is whether or not it is invertible. Think about the Invertible Matrix Theorem and all the

Solution

given that 0 is an eigenvalue. Thus there is some nontrivial solution to Ax = 0x = 0. By the invertible matrix

theorem, if A was invertible there would only be the trivial solution. Since there is a nontrivial solution, it must be the case that A is NOT INVERTIBLE.