Matrices & Determinants
Grade 12
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.
B = A + I, C = -I
C = A + I, B = -I
B = A - I, C = -I
B = A + I, C = +I
Step-by-Step Solution
Key Concept: Derive the characteristic equation of matrix A and algebraically manipulate the resulting polynomial identity to isolate A as a sum of two cubes.
<div>$A^2 = \begin{bmatrix} -1 & 0 \\ k & -2 \end{bmatrix} \begin{bmatrix} -1 & 0 \\ k & -2 \end{bmatrix} \Rightarrow A^2 = \begin{bmatrix} 1 & 0 \\ -3k & 4 \end{bmatrix}$<br/>$\Rightarrow A^2 + 3A = \begin{bmatrix} 1 & 0 \\ -3k & 4 \end{bmatrix} + \begin{bmatrix} -3 & 0 \\ 3k & -6 \end{bmatrix} = \begin{bmatrix} -2 & 0 \\ 0 & -2 \end{bmatrix}$<br/>$\Rightarrow A^2 + 3A + 2I = \begin{bmatrix} -2 & 0 \\ 0 & -2 \end{bmatrix} + \begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix} = \mathbf{0}$.<br/>$\therefore A^3 + 3A^2 + 2A = 0$.<br/>$\Rightarrow (A + I)^3 - A = I^3 \Rightarrow A = (A + I)^3 + (-I)^3$<br/>$\Rightarrow B^3 + C^3 \Rightarrow B = A + I$ and $C = -I$</div>
Correct Answer: A