User:Thenewandyl/MA 3113 Test 2 Question 4

We had a matrix, A, defined as:

$$ A = \begin{bmatrix} 2^{1/3} & 0 \\ 0 & 3^{1/3} \end{bmatrix} $$

B was defined as something funky, but it really didn't matter what B was, only that it was defined as:

$$ B = P^{-1}AP\, $$

It was necessary to realize that this pattern applied:

$$ B^2 = {(P^{-1}AP)}^2 = {(P^{-1}AP)}{(P^{-1}AP)} = P^{-1}A(PP^{-1})AP = P^{-1}AAP = P^{-1}A^2P $$

In fact, this holds for Bn:

$$ B^n = P^{-1}A^nP\, $$

Note that since A is a diagonal matrix, this property holds:

$$ A^n = \begin{bmatrix} {a_{11}}^n & 0 \\ 0 & {a_{22}}^n \end{bmatrix} $$

So:

$$ A^6 = \begin{bmatrix} {(2^{1/3})}^6 & 0 \\ 0 & {(3^{1/3})}^6 \end{bmatrix} = \begin{bmatrix} 4 & 0 \\ 0 & 9 \end{bmatrix} $$

So, I don't remember the exact values of P and P-1, but the equation in the end was:

$$ B^6 = P^{-1}A^6P = P^{-1} \begin{bmatrix} 4 & 0 \\ 0 & 9 \end{bmatrix} P = \begin{bmatrix} 9 & 0 \\ 5 & 4 \end{bmatrix} $$

Again, that's if my memory serves me correctly.