1. ## Matrices

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 have to do this question without a calculator.

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?

2. Originally Posted by Showcase_22
I have to do this question without a calculator.
But have you tried the question with a calculator?
Had you done that you would have see some direction.

For one, the problem as posted has a determinate of 0.
That cannot be correct. After looking at your work, it is clear that $\displaystyle a_{2,1}=1$ not 2 as you posted.
Then the matrix has determinate equal 2.