Two matrices A B MnR are called similar if B SAS1 for some

Two matrices A, B Mn(R) are called similar if B = SAS1 for some invertible matrix S GLn(R).

Question 1. Prove the following:

• Every square matrix A is similar to itself.

• If A is similar to B, then B is similar to A.

• If A is similar to B, and B is similar to C, then A is similar to C.

Solution

1> I = identity matrix which belongs to GLn(R).

I^-1= I

Now IAI^-1= AI (since IA= A and I^-1=I)

= A

so A is similar to itself.

A is similar to B

=> B = SAS^-1 for some S belongs to GLn(R)

=> BS = SAS^-1S

=> BS = SA since S^-1S= I and IA=A

=> S^-1BS=S^-1SA   

=> S^-1BS = A since S^-1S= I and IA=A

=> A = S^-1B(S^-1)^-1   since S=(S^-1)^-1

=> B is similar to A