FP1 January 2011 Q2
2. \[\mathbf{A} = \begin{pmatrix} 2 & 0 \\ 5 & 3 \end{pmatrix}, \quad \mathbf{B} = \begin{pmatrix} -3 & -1 \\ 5 & 2 \end{pmatrix}\]
(a) Find \(\mathbf{AB}\). (3)
Given that \[\mathbf{C} = \begin{pmatrix} -1 & 0 \\ 0 & 1 \end{pmatrix}\]
(b) describe fully the geometrical transformation represented by \(\mathbf{C}\), (2)
(c) write down \(\mathbf{C}^{100}\). (1)
| Scheme | Marks |
|---|---|
| \(\mathbf{A} = \begin{pmatrix} 2 & 0 \\ 5 & 3 \end{pmatrix}, \mathbf{B} = \begin{pmatrix} -3 & -1 \\ 5 & 2 \end{pmatrix}\) \(\mathbf{AB} = \begin{pmatrix} 2 & 0 \\ 5 & 3 \end{pmatrix}\begin{pmatrix} -3 & -1 \\ 5 & 2 \end{pmatrix}\) | |
| \(= \begin{pmatrix} 2(-3) + 0(5) & 2(-1) + 0(2) \\ 5(-3) + 3(5) & 5(-1) + 3(2) \end{pmatrix}\) A correct method to multiply out two matrices. Can be implied by two out of four correct elements. | M1 |
| \(= \begin{pmatrix} -6 & -2 \\ 0 & 1 \end{pmatrix}\) Any three elements correct | A1 |
| Correct answer Correct answer only 3/3 | A1 |
| (3) |
| Scheme | Marks |
|---|---|
| Reflection; about the \(y\)-axis. Reflection \(y\)-axis (or \(x = 0\).) | M1 A1 |
| (2) |
| Scheme | Marks |
|---|---|
| \(\mathbf{C}^{100} = \mathbf{I} = \underline{\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}}\) \(\underline{\begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}}\) or \(\mathbf{I}\) | B1 |
| (1) | |
| [6] |