3 Find a matrix \(\mathbf { A }\) whose eigenvalues are \(- 1,1,2\) and for which corresponding eigenvectors are
$$\left( \begin{array} { l }
1 \\
0 \\
0
\end{array} \right) , \quad \left( \begin{array} { l }
1 \\
1 \\
0
\end{array} \right) , \quad \left( \begin{array} { l }
0 \\
1 \\
1
\end{array} \right) ,$$
respectively.
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Question 3:
Answer Marks
Guidance
Answer/Working Marks
Guidance
\(\mathbf{P} = \begin{pmatrix}1&1&0\\0&1&1\\0&0&1\end{pmatrix}\), \(\mathbf{D} = \begin{pmatrix}-1&0&0\\0&1&0\\0&0&2\end{pmatrix}\) B1B1
soi
\(\mathbf{P}^{-1}\mathbf{A}\mathbf{P} = \mathbf{D} \Rightarrow \mathbf{A} = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}\) M1
soi
\(\mathbf{P}^{-1} = \begin{pmatrix}1&-1&1\\0&1&-1\\0&0&1\end{pmatrix}\) M1A1
\(\mathbf{PD} = \begin{pmatrix}-1&1&0\\0&1&2\\0&0&2\end{pmatrix}\) or \(\mathbf{DP}^{-1} = \begin{pmatrix}-1&1&-1\\0&1&-1\\0&0&2\end{pmatrix}\) M1
\(\mathbf{A} = \begin{pmatrix}-1&2&-2\\0&1&1\\0&0&2\end{pmatrix}\) A1
[7]
ALT: \(Av_1=-v_1,\ Av_2=v_2,\ Av_3=2v_3\) M1, A1 ; multiply out, solve each equation M1, A1, A1, A1 ; \(\mathbf{A}=\begin{pmatrix}-1&2&-2\\0&1&1\\0&0&2\end{pmatrix}\) B1 \(\checkmark\)
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## Question 3:
| Answer/Working | Marks | Guidance |
|---|---|---|
| $\mathbf{P} = \begin{pmatrix}1&1&0\\0&1&1\\0&0&1\end{pmatrix}$, $\mathbf{D} = \begin{pmatrix}-1&0&0\\0&1&0\\0&0&2\end{pmatrix}$ | B1B1 | soi |
| $\mathbf{P}^{-1}\mathbf{A}\mathbf{P} = \mathbf{D} \Rightarrow \mathbf{A} = \mathbf{P}\mathbf{D}\mathbf{P}^{-1}$ | M1 | soi |
| $\mathbf{P}^{-1} = \begin{pmatrix}1&-1&1\\0&1&-1\\0&0&1\end{pmatrix}$ | M1A1 | |
| $\mathbf{PD} = \begin{pmatrix}-1&1&0\\0&1&2\\0&0&2\end{pmatrix}$ or $\mathbf{DP}^{-1} = \begin{pmatrix}-1&1&-1\\0&1&-1\\0&0&2\end{pmatrix}$ | M1 | |
| $\mathbf{A} = \begin{pmatrix}-1&2&-2\\0&1&1\\0&0&2\end{pmatrix}$ | A1 | [7] |
| ALT: $Av_1=-v_1,\ Av_2=v_2,\ Av_3=2v_3$ **M1, A1**; multiply out, solve each equation **M1, A1, A1, A1**; $\mathbf{A}=\begin{pmatrix}-1&2&-2\\0&1&1\\0&0&2\end{pmatrix}$ **B1**$\checkmark$ | | |
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3 Find a matrix $\mathbf { A }$ whose eigenvalues are $- 1,1,2$ and for which corresponding eigenvectors are
$$\left( \begin{array} { l }
1 \\
0 \\
0
\end{array} \right) , \quad \left( \begin{array} { l }
1 \\
1 \\
0
\end{array} \right) , \quad \left( \begin{array} { l }
0 \\
1 \\
1
\end{array} \right) ,$$
respectively.
\hfill \mbox{\textit{CAIE FP1 2016 Q3 [7]}}