Question

A is an n×n matrix. Check the true statements below: A. Finding an eigenvector of A might be difficult, but checking whether a given vector is in fact an eigenvector is easy. B. A matrix A is not invertible if and only if 0 is an eigenvalue of A. C. To find the eigenvalues of A, reduce A to echelon form. D. A number c is an eigenvalue of A if and only if the equation (A−c????)x=0 has a nontrivial solution x. E. If Ax=????x for some vector x, then ???? is an eigenvalue of A.

Answer 1

A,B,D

Explanation:

Answer 2

???? = Identity matrix

???? = c

Explanation:

Remember that you are looking for a value "c" and a vector "x" such that

Ax = cx

In that case "x" is an eigenvector and "c" is an eigenvalue. Therefore if you subtract "cx" from both sides of the equality you have that

Ax-cx = 0 , and ( A - Ic )x = 0 , where " I " is the identity matrix. And "c" is the eigenvalue.