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.

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- May 4th 2011, 06:32 AMkaragorgeProblem with a matrix
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. - May 4th 2011, 07:01 AMTheEmptySet
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

$\displaystyle \begin{bmatrix} -2 & 0 \\ 0 & -2\end{bmatrix}$

$\displaystyle \begin{bmatrix} -2 & 1 \\ 0 & -2\end{bmatrix}$

$\displaystyle \begin{bmatrix} 2 & 0 \\ 0 & 2\end{bmatrix}$

$\displaystyle \begin{bmatrix} -2 & 0 \\ 0 & 2\end{bmatrix}$