1. ## Theoretical (Similar Matrix)

Show that:
A) A is similar to A.
B) If B is similar to A, then A is similar to B.
C) If A is similar to B and B is similar to C, then A is similar to C.

I dont know how to do the theoretical problems in my book. I know that similar means:

A matrix B is said to be similar to a matrix A if there is a nonsingular matrix P such that
B = P^-1 A P

2. Originally Posted by PensFan10
Show that:
A) A is similar to A.
B) If B is similar to A, then A is similar to B.
C) If A is similar to B and B is similar to C, then A is similar to C.

I dont know how to do the theoretical problems in my book. I know that similar means:

A matrix B is said to be similar to a matrix A if there is a nonsingular matrix P such that
B = P^-1 A P

A is similar to matrix say B iff there exist an invertible matrix p such that

$\displaystyle P^{-1} A P = B$

so since the inverse for the identity is the identity so

$\displaystyle I^{-1} A I = A$ this is true we find the matrix p

ok second one
B is similar to A so there exist p such that

$\displaystyle P^{-1} B P = A$

$\displaystyle P\cdot P^{-1} B P = P A$

$\displaystyle B P\cdot P^{-1} = P A \cdot P^{-1}$

$\displaystyle B=P A P^{-1}$

let $\displaystyle P^{-1} = S$

$\displaystyle B = S^{-1} A S$ we find the matrix so A is similar to B try in the last one

3. So for the last one:

B = R^-1 C R A = P^-1 B P

wth substitution I get
A = P^-1(R^-1 C R) P

not sure where to go from here. I dont know how to simplify this to get that A is similar to C

4. Originally Posted by PensFan10
So for the last one:

B = R^-1 C R A = P^-1 B P

wth substitution I get
A = P^-1(R^-1 C R) P

not sure where to go from here. I dont know how to simplify this to get that A is similar to C
in general
the inverse for the matrix

$\displaystyle AB$

is $\displaystyle B^{-1}A^{-1}$

since

$\displaystyle AB \cdot B^{-1}A^{-1} = I$

$\displaystyle RP = S$
$\displaystyle S^{-1}=$