1. ## Matrices Problem?

So I'm trying to solve this sample question because I'll be required to solve a similar one for an assignment but I'm not sure how to go about it.

I know that A(^2) is (11, 28 ; 4, 11) and A(^4) is (233, 616 ; 88, 233)but I'm not sure how to find m and n, can anyone help? I don't even know if I can add a constant to a matrix.
I'm more interested in how the question is done than the answer so a step by step would be appreciated!

2. ## Re: Matrices Problem?

Originally Posted by Daithi19
So I'm trying to solve this sample question because I'll be required to solve a similar one for an assignment but I'm not sure how to go about it.

I know that A(^2) is (11, 28 ; 4, 11) and A(^4) is (233, 616 ; 88, 233)but I'm not sure how to find m and n, can anyone help? I don't even know if I can add a constant to a matrix.
I'm more interested in how the question is done than the answer so a step by step would be appreciated!
So substitute these into your equation

\displaystyle \displaystyle \begin{align*} \mathbf{A}^4 &= m\mathbf{A}^2 + n\mathbf{I}_2 \\ \left[\begin{matrix} 233 & 616 \\ 88 & 233 \end{matrix}\right] &= m\left[\begin{matrix} 11 & 28 \\ 4 & 11 \end{matrix}\right] + n\left[\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}\right] \\ \left[\begin{matrix} 233 & 616 \\ 88 & 233 \end{matrix}\right] &= \left[\begin{matrix} 11m + n & 28m \\ 4m & 11m + n \end{matrix}\right] \end{align*}

So from here you can see

\displaystyle \displaystyle \begin{align*} 233 &= 11m + n \\ 616 &= 28m \\ 88 &= 4m \end{align*}

Solve this for \displaystyle \displaystyle \begin{align*} m \end{align*} and \displaystyle \displaystyle \begin{align*} n \end{align*}

3. ## Re: Matrices Problem?

Ok so m = 22 and n = -9 , but do you mind explaining where you got the matrix that you multiplied by N from?

4. ## Re: Matrices Problem?

Originally Posted by Daithi19
Ok so m = 22 and n = -9 , but do you mind explaining where you got the matrix that you multiplied by N from?
You can't add scalars with matrices, so I multiplied the scalar by the identity matrix. This is a standard technique with matrix polynomials.