Matrices & Determinants Question - None

Question

Determine the matrix $B$ and $C$ with integral element such that $A = \begin{bmatrix} -1 & 0 \ k & -2 \end{bmatrix} = B^3 + C^3$, where $k$ is any integer.

Accepted Answer

$A^2 = \begin{bmatrix} -1 & 0 \ k & -2 \end{bmatrix} \begin{bmatrix} -1 & 0 \ k & -2 \end{bmatrix} \Rightarrow A^2 = \begin{bmatrix} 1 & 0 \ -3k & 4 \end{bmatrix}$
$\Rightarrow A^2 + 3A = \begin{bmatrix} 1 & 0 \ -3k & 4 \end{bmatrix} + \begin{bmatrix} -3 & 0 \ 3k & -6 \end{bmatrix} = \begin{bmatrix} -2 & 0 \ 0 & -2 \end{bmatrix}$
$\Rightarrow A^2 + 3A + 2I = \begin{bmatrix} -2 & 0 \ 0 & -2 \end{bmatrix} + \begin{bmatrix} 2 & 0 \ 0 & 2 \end{bmatrix} = \begin{bmatrix} 0 & 0 \ 0 & 0 \end{bmatrix} = \mathbf{0}$.
$ herefore A^3 + 3A^2 + 2A = 0$.
$\Rightarrow (A + I)^3 - A = I^3 \Rightarrow A = (A + I)^3 + (-I)^3$
$\Rightarrow B^3 + C^3 \Rightarrow B = A + I$ and $C = -I$

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