Matrices that commute with a given matrix
Matrices that commute with a given matrix
IP 3
9 marks
\(\mathbf{A}=\begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}\).
- Find a \(2\times2\) matrix \(\mathbf{B}\) for which \(\mathbf{AB}\ne\mathbf{BA}\).[2]
- Find every matrix \(\mathbf{B}=\begin{pmatrix} p & q \\ r & s \end{pmatrix}\) for which \(\mathbf{AB}=\mathbf{BA}\).[4]
- Find \(\mathbf{A}^2\) and \(\mathbf{A}^3\), and hence suggest a general expression for \(\mathbf{A}^n\), where \(n\) is a positive integer. Explain why this agrees with your answer to (b).[3]
Solution:
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Answer:(a) e.g. \(\mathbf{B}=\begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix}\) (b) \(\mathbf{B}=\begin{pmatrix} p & q \\ 0 & p \end{pmatrix}\) (c) \(\mathbf{A}^n=\begin{pmatrix} 1 & 2n \\ 0 & 1 \end{pmatrix}\)
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