1. ## Polynomial

|-1 0 0 0|
|0 -1 3 0|
|0 0 2 0|
|-3 3 -3 2|

Given this matrix A, and the polynom f(x) = x^2 + x + 2, what is the matrix M = f(A) ? Thanks alot!

2. Originally Posted by gralla55
|-1 0 0 0|
|0 -1 3 0|
|0 0 2 0|
|-3 3 -3 2|

Given this matrix A, and the polynom f(x) = x^2 + x + 2, what is the matrix M = f(A) ? Thanks alot!

$\displaystyle \left[\begin{array}{cccc}-1&0&0&0\\0&-1&3&0\\0&0&2&0\\-3&3&-3&2\end{array}\right]$

$\displaystyle f(x)= x^2 + x +2$

$\displaystyle f(A) = A^2 + A + 2I$ I is the identity

$\displaystyle f(A) =\left[\begin{array}{cccc}-1&0&0&0\\0&-1&3&0\\0&0&2&0\\-3&3&-3&2\end{array}\right]\times \left[\begin{array}{cccc}-1&0&0&0\\0&-1&3&0\\0&0&2&0\\-3&3&-3&2\end{array}\right]$$\displaystyle + \left[\begin{array}{cccc}-1&0&0&0\\0&-1&3&0\\0&0&2&0\\-3&3&-3&2\end{array}\right] + 2 \left[\begin{array}{cccc}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0& 0&1\end{array}\right]$

3. Thanks alot! If I did it right, the squared matrix should be:

1 0 0 0
0 1 3 0
0 0 4 0
-3 3 -3 4

which makes the final matrix:

2 0 0 0
0 2 6 0
0 0 8 0
-6 6 -6 8

Does this look correct?

4. it is correct