Hi i got stuck on this problem - pls help if u can:
Find all scalar matrices
A^3 + 2A^2 - 4A - 8I_2 = 0
where I_2 is identity matrix.
Given the above get the polynomial
$\displaystyle x^3+2x^2-4x-8=0 \iff x^2(x+2)-4(x+2)=0 \iff (x+2)^2(x-2)=0$
Since this polynomial is in the annihilating Ideal the minimum polynomial must divide the above polynomial
Since we are working with
$\displaystyle 2 \times 2$
Matrices
We can have the following minimum polynomials
$\displaystyle (x+2);\quad (x+2)^2; \quad (x-2); \quad (x+2)(x-2)$
So up to similarity you will get the 4 matrices
$\displaystyle \begin{bmatrix} -2 & 0 \\ 0 & -2\end{bmatrix}$
$\displaystyle \begin{bmatrix} -2 & 1 \\ 0 & -2\end{bmatrix}$
$\displaystyle \begin{bmatrix} 2 & 0 \\ 0 & 2\end{bmatrix}$
$\displaystyle \begin{bmatrix} -2 & 0 \\ 0 & 2\end{bmatrix}$