Thread: Last Linear Algebra problem

Defenition: Let A and B be nxn matrices. Then A is said to be similar to B, denoted A is congruent to B, if there exists an invertible nxn matrix P such that B=(P^-1)AP.

a. Let A=I_n. Prove that, if A is congruent to B, then B=I_n

b. Let A be a nxn idempotent matrix. Prove that if A is congruent to B, then B is idempotent.

c. Is the analogue of part b above the nilpotent matrices true? Why or Why not?

Originally Posted by natewalker205

Defenition: Let A and B be nxn matrices. Then A is said to be similar to B, denoted A is congruent to B, if there exists an invertible nxn matrix P such that B=(P^-1)AP.

a. Let A=I_n. Prove that, if A is congruent to B, then B=I_n

b. Let A be a nxn idempotent matrix. Prove that if A is congruent to B, then B is idempotent.

c. Is the analogue of part b above the nilpotent matrices true? Why or Why not?

In all of these cases you just need to observe that as and are congruent there exists an invertible matrix such that:



Now use what you are told in a, b and c and the answers follow at once

RonL