Question

if A and B are similar matrices, prove that A^2 and B^2 are also similar matrices

Answer 1

does this work? A = P^-1*B*P

A*A = p^-1*B*p*p-1*B*p
A*A = p^-1*B*B*p
A^2 = P^-1*B^2*P

Answer 2

What defines a "similar matrix"?

Answer 3

that first line, where \[A=p^{-1}*B*p\]

Answer 4

What does \(p\) represent?

Answer 5

Thats perfect :)
The definition of a matrix A being similar to a matrix B is that there exists an invertible matrix P such that:\[A=PBP^{-1}\]or the other way around with the inverse, it doesnt matter. So we can show A^2 is similar to B^2 by doing exactly what Junkie did:\[A^2=AA=(PBP^{-1})(PBP^{-1})=PB(P^{-1}P)BP^{-1}=PB(I)BP^{-1}\]\[PBBP^{-1}=PB^2P^{-1}\]