1.) Given two matrices, A and B, show that

A = [[1,1],[-1,4]], and

B =[[2,1],[1,3]] are similar.

Then, show that the matrix

C = [[3,1],[-6,-2]] and

D = [[-1,2],[1,0]] are NOT similar.

Could you check to see if I am right? The theorem for similar matrices states that if n x n matrices A and B are similar, then they have the same characteristic polynomial and hence the same eigenvalues (with the same multiplicities).

Or: Two n × n matrices A and B are similar, if there exists a non-singular n × n matrix P such that B = P−1AP.

For A,

(1*4) - (-1*1) = 5, so it is non-singular.

And B,

(2*3) - (1*1) = 5, so it is non-singular

(A - LI)x = 0

A = [[1-L, 1],[-1,4-L]]

(1-L)*(4-L) - (-1) = 0

Solving for L, we get the eigenvalues are 5/2+1/2*sqrt(5), 5/2-1/2*sqrt(5)

B = [[2-L, 1],[1, 3-L]]

(2-L)*(3-L) - 1*1 = 0

Solving for L, we get the eigenvalues are the same as above.

Is this sufficient to show that they are similar, since they both have the same eigenvalues?

To show C and D are not similar:

For C:

(3-L)*(-2-L) - (-6) = 0

Solving for L, we get the eigenvalues are 0 and 1.

For D:

(-1-L)*(0-L) - 2 = 0

Solving for L, we get the eigenvalues are 1 and -2.

Is this sufficient to prove that they are not similar, since they don't have the same eigenvalues?