Non-zero matrices whose square is zero
Non-zero matrices whose square is zero
IP 3
8 marks
- Show that \(\begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}^2=\mathbf{0}\). What does this show about solving \(\mathbf{A}^2=\mathbf{0}\)?[2]
- Show that for any \(\mathbf{A}=\begin{pmatrix} a & b \\ c & -a \end{pmatrix}\), \(\mathbf{A}^2=(a^2+bc)\mathbf{I}\).[2]
- Hence find a matrix \(\mathbf{A}\), with every element non-zero, such that \(\mathbf{A}^2=\mathbf{0}\).[2]
- Explain why a matrix \(\mathbf{A}\) with \(\mathbf{A}^2=\mathbf{0}\) cannot have an inverse.[2]
Solution:
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Answer:(c) e.g. \(\begin{pmatrix} 2 & 4 \\ -1 & -2 \end{pmatrix}\)
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