c) Suppose that A is an n × n matrix, and that the characteristic polynomial of A
had n distinct real roots. Explain why A is similar to a diagonal matrix.

d) Suppose that A and B are two n × n matrices with the same characteristic poly-
nomial, and that the characteristic polynomial has n distinct real roots (i.e., like
part (c)). A must be similar to B.)

(e) Find an example of 2 × 2 matrices A and B which have the same characteristic
polynomial, but are not similar. (For instance, perhaps one is diagonalizable, and
one is not.)

Question

c) Suppose that A is an n × n matrix, and that the characteristic polynomial of A
had n distinct real roots. Explain why A is similar to a diagonal matrix. 

d) Suppose that A and B are two n × n matrices with the same characteristic poly-
nomial, and that the characteristic polynomial has n distinct real roots (i.e., like
part (c)). A must be similar to B.)

(e) Find an example of 2 × 2 matrices A and B which have the same characteristic
polynomial, but are not similar. (For instance, perhaps one is diagonalizable, and
one is not.)