Problem with a matrix

**karagorge** · May 4th 2011, 06:32 AM

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.

**TheEmptySet** · May 4th 2011, 07:01 AM

Originally Posted by karagorge

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

$x^3+2x^2-4x-8=0 \iff x^2(x+2)-4(x+2)=0 \iff (x+2)^2(x-2)=0$

Since this polynomial is in the annihilating Ideal the minimum polynomial must divide the above polynomial

Since we are working with

$2 \times 2$

Matrices
We can have the following minimum polynomials
$(x+2);\quad (x+2)^2; \quad (x-2); \quad (x+2)(x-2)$

So up to similarity you will get the 4 matrices

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

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

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

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

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