Let 2 be an eigenvalue of an invertible matrix A. Show that A-1 is an eigenvalue of A -1 [Hint: Suppose a nonzero satisfies Ax =x.] ..... -1 exists. In order for A to be an eigenvalue of A', there...

I'm confused on what equation they are asking for?

Extracted text: Let 2 be an eigenvalue of an invertible matrix A. Show that A-1 is an eigenvalue of A -1 [Hint: Suppose a nonzero satisfies Ax =x.] ..... -1 exists. In order for A to be an eigenvalue of A', there must exist a nonzero x such that A 1x =1-1x. -1 Note that A Suppose a nonzero satisfies Ax = Ax. What is the first operation that should be performed on Ax = x so that an equation similar to the one in the previous step can be obtained? O A. Invert the product on each side of the equation. O B. Left-multiply both sides of Ax = x by A1. Oc. Right-multiply both sides of Ax = ix by A1. Perform the operation and simplify. |(Type an equation. Simplify your answer.) Why does this show that 11 is defined? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. O A. Since x is an eigenvector of A, A -1 and x are commutable. By definition, x is nonzero, so the previous equation cannot be satisfied if A = O B. By definition, x is nonzero and A is invertible. So, the previous equation cannot be satisfied if ) = 0. O C. Since the product 2-1x must be defined and nonzero, A-1 must exist and be nonzero. How does this show that 1 is an eigenvalue of A? Select the correct choice below and fill in the answer box to complete your choice. (Simplify your answer.) O A. Both sides of the equation can be multiplied by and one side can be simplified to obtain (A-2-0x= 0. O B. Both sides of the equation can be multiplied by and one side can be simplified to obtain 21A-x= 0. OC. Both sides of the equation can be multiplied by A and one side can be simplified to obtain A1x = Ax. O D. Both sides of the equation can be multiplied by and one side can be simplified to obtain 21AX=x.