Determinant of a product and cancelling a matrix
Determinant of a product and cancelling a matrix
IP 3
8 marks
\(\mathbf{A}=\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) and \(\mathbf{B}=\begin{pmatrix} p & q \\ r & s \end{pmatrix}\) are any \(2\times2\) matrices.
- Show that \(\det(\mathbf{AB})=\det(\mathbf{A})\det(\mathbf{B})\).[4]
- Given that \(\det(\mathbf{A})=3\), find \(\det(\mathbf{A}^3)\) and explain why \(\mathbf{A}^3\) has an inverse.[2]
- By considering \(\mathbf{C}=\begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix}\), show that \(\mathbf{CX}=\mathbf{CY}\) does not always imply \(\mathbf{X}=\mathbf{Y}\).[2]
Solution:
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Answer:(b) \(27\) (c) e.g. different \(\mathbf{X}\), \(\mathbf{Y}\) give \(\mathbf{CX}=\mathbf{CY}\) since \(\det(\mathbf{C})=0\)
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