FP1 June 2017 Q2
2. \[\mathbf{A} = \begin{pmatrix} 2 & -1 \\ 4 & 3 \end{pmatrix}, \quad \mathbf{P} = \begin{pmatrix} 3 & 6 \\ 11 & -8 \end{pmatrix}\]
The transformation represented by the matrix B followed by the transformation represented by the matrix A is equivalent to the transformation represented by the matrix P.
| Scheme | Marks |
|---|---|
| \(\mathbf{A} = \begin{pmatrix} 2 & -1 \\ 4 & 3 \end{pmatrix},\ \mathbf{P} = \begin{pmatrix} 3 & 6 \\ 11 & -8 \end{pmatrix}\) | |
| \(\mathbf{A}^{-1} = \dfrac{1}{10}\begin{pmatrix} 3 & 1 \\ -4 & 2 \end{pmatrix}\) | M1 A1 |
| (2) |
Notes
M1: Either \(\dfrac{1}{10}\) or \(\begin{pmatrix} 3 & 1 \\ -4 & 2 \end{pmatrix}\)
A1: Correct matrix seen.
| Scheme | Marks |
|---|---|
| Way 1 \(\mathbf{P} = \mathbf{AB}\) \(\Rightarrow \mathbf{A}^{-1}\mathbf{P} = \mathbf{A}^{-1}\mathbf{AB} \Rightarrow \mathbf{B} = \mathbf{A}^{-1}\mathbf{P}\) | |
| \(\mathbf{B} = \dfrac{1}{10}\begin{pmatrix} 3 & 1 \\ -4 & 2 \end{pmatrix}\begin{pmatrix} 3 & 6 \\ 11 & -8 \end{pmatrix}\) | M1 |
| \(= \begin{pmatrix} 2 & 1 \\ 1 & -4 \end{pmatrix}\) | A1 A1 |
| (3) | |
| (5 marks) |
Notes
M1: Multiplies their \(\mathbf{A}^{-1}\) by P in correct order. This substituted statement is sufficient.
A1: At least 2 elements correct or \(k\begin{pmatrix} 20 & 10 \\ 10 & -40 \end{pmatrix}\) oe. May be unsimplified
A1: Correct simplified matrix.
(b) Way 2
| Scheme | Marks |
|---|---|
| \(\left\{\mathbf{P} = \mathbf{AB} \Rightarrow\right\}\) \(\begin{pmatrix} 3 & 6 \\ 11 & -8 \end{pmatrix} = \begin{pmatrix} 2 & -1 \\ 4 & 3 \end{pmatrix}\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) | |
| \(\begin{pmatrix} 3 & 6 \\ 11 & -8 \end{pmatrix} = \begin{pmatrix} 2a - c & 2b - d \\ 4a + 3c & 4b + 3d \end{pmatrix}\) | M1 |
| \(\Rightarrow a = 2,\ c = 1,\ b = 1,\ d = -4\) | |
| So, \(\mathbf{B} = \begin{pmatrix} 2 & 1 \\ 1 & -4 \end{pmatrix}\) | A1 A1 |
| (3) |
M1: Attempt to multiply A by B in the correct order and puts equal to P
A1: At least 2 elements are correct.
A1: Correct matrix.