Similar matrices

1. The problem statement, all variables and given/known data
1) Show that for any real numbers a and b, the matrices

[tex] \left( \begin{array}{cc}
1 & a \\
0 & 2 \\
\end{array}\right)[/tex]

and

[tex] \left(\begin{array}{cc}
1 & b \\
0 & 2 \\
\end{array}\right)[/tex]

are similar.

2) Show that

[tex] \left( \begin{array}{cc}
2 & 1 \\
0 & 2 \\
\end{array}\right)[/tex]

and

[tex] \left(\begin{array}{cc}
2 & 0 \\
0 & 2 \\
\end{array}\right)[/tex]

are not similar.
2. Relevant equations



3. The attempt at a solution

We want to show that B=P^{-1} A P holds for some P in the first case and holds for no P in the second case. So I let P be an arbitrary 2 by 2 matrix and just wrote out the four equations that you get using the explicit formula for the inverse but that failed . So what is the trick...

 

anyone?

 

OK. Here is the solution to part 2: Let
P = [tex]
\left( \begin{array}{cc}
w & x \\
y & z \\
\end{array}\right)
[/tex]

Then, if we write equations for each element of AP=PB, where A is the first matrix and B is the second matrix, we see that w=y=0 and a matrix with a column of zeros is clearly not invertible.

We can do the same thing for part 1) and obtain PA=BP (where A is the first matrix and B is the second one) if x=y=0 and w=b and z=a. Then P is invertible whenever a or b are not zero. If they are both zero, this is trivial. If a is zero and b is nonzero we let x=b,z=1, and w= anything nonzero, y = 0. And then when a is nonzero and b is zero do the "same" thing.