Generalize the idea of $A^2 = I_2$ to find a $5 \times 5$ matrix such that $M^2 = 0$ and $C$ is a nonzero $2 \times 3$ matrix

Question

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I know how to solve part (a) and (b), but I'm having trouble understanding question (c). In part (a) the A matrix is $2 \times 2$ but in question (c) it is $3 \times 3$ to satisfy the $M$ matrix dimension. Can anybody explain how I can approach (c)?

Answer 1

We must have $A \in M_{3 \times 3}(\mathbb{R}), C \in M_{2 \times 3}(\mathbb{R})$ and $D \in M_{2 \times 2}(\mathbb{R})$. You have

$$ M^2 = \begin{pmatrix} A & 0 \\ C & D \end{pmatrix} \begin{pmatrix} A & 0 \\ C & D \end{pmatrix} = \begin{pmatrix} A^2 & 0 \\ CA + DC & D^2 \end{pmatrix}. $$

I'm not sure what the author means by the hint, but if you take $A = D = 0$ and $C$ any non-zero matrix, you will have $M^2 = 0$.