I have to do this question without a calculator.Quote:

Let A= $\displaystyle \begin{pmatrix}

{4}&{2}\\

{1}&{1}

\end{pmatrix}$. Then what is the value of $\displaystyle Det(A^{11}-A^{10})$?

a) -17

b) $\displaystyle -2^9$ x 17

c) $\displaystyle -2^{11}$

d) -24

I could work it out doing lots of multiplication but this question is taken off a test where we have to do 11 questions in an hour. There must be a shorter way of doing it.

Here was another way I was thinking of:

$\displaystyle Det(M- \lambda I)=\begin{vmatrix}

{4-\lambda}&{2}\\

{1}&{1-\lambda}

\end{vmatrix}=(4-\lambda)(1-\lambda)-2=\lambda^2-5 \lambda+2$

The disciminant is $\displaystyle \sqrt{25-8}=\sqrt 17$

I can see that the 17 from the choices is appearing but since it's $\displaystyle A^{11}$ then i'm going to get a $\displaystyle \sqrt {17}$ in my answer somewhere.

Can anyone help?