A matrix that satisfies its own quadratic equation
A matrix that satisfies its own quadratic equation
IP 3
8 marks
\(\mathbf{A}=\begin{pmatrix} x & 2 \\ 1 & x \end{pmatrix}\), where \(x\) is a constant, and \(\mathbf{I}\) is the \(2\times2\) identity matrix.
- Find the value of \(x\) for which \(\mathbf{A}^2-4\mathbf{A}\) is a multiple of \(\mathbf{I}\), and state this multiple.[3]
- For this value of \(x\), use your result to show that \(\mathbf{A}^{-1}=\dfrac{1}{2}(4\mathbf{I}-\mathbf{A})\), and write down \(\mathbf{A}^{-1}\).[3]
- Hence solve the simultaneous equations \(2u+2v=6\) and \(u+2v=5\).[2]
Solution:
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Answer:(a) \(x=2\); \(\mathbf{A}^2-4\mathbf{A}=-2\mathbf{I}\) (b) \(\mathbf{A}^{-1}=\begin{pmatrix} 1 & -1 \\ -\dfrac{1}{2} & 1 \end{pmatrix}\) (c) \(u=1,\ v=2\)
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