Date | May 2018 | Marks available | 2 | Reference code | 18M.1.hl.TZ0.13 |
Level | HL only | Paper | 1 | Time zone | TZ0 |
Command term | State | Question number | 13 | Adapted from | N/A |
Question
Consider the matrix M = \(\left[ {\begin{array}{*{20}{c}}
2 \\
{ - 1}
\end{array}\,\,\,\begin{array}{*{20}{c}}
{ - 4} \\
{ - 1}
\end{array}} \right]\).
Show that the linear transformation represented by M transforms any point on the line \(y = x\) to a point on the same line.
Explain what happens to points on the line \(4y + x = 0\) when they are transformed by M.
State the two eigenvalues of M.
State two eigenvectors of M which correspond to the two eigenvalues.
Markscheme
\(\left( {\begin{array}{*{20}{c}}
2 \\
{ - 1}
\end{array}\,\,\,\begin{array}{*{20}{c}}
{ - 4} \\
{ - 1}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
k \\
k
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
{ - 2k} \\
{ - 2k}
\end{array}} \right)\left( { = - 2\left( {\begin{array}{*{20}{c}}
k \\
k
\end{array}} \right)} \right)\) M1A1
hence still on the line \(y = x\) AG
[2 marks]
consider \(\left( {\begin{array}{*{20}{c}}
2 \\
{ - 1}
\end{array}\,\,\,\begin{array}{*{20}{c}}
{ - 4} \\
{ - 1}
\end{array}} \right)\left( {\begin{array}{*{20}{c}}
{4k} \\
{ - k}
\end{array}} \right)\) M1
\( = \left( {\begin{array}{*{20}{c}}
{12k} \\
{ - 3k}
\end{array}} \right)\left( { = 3\left( {\begin{array}{*{20}{c}}
{4k} \\
{ - k}
\end{array}} \right)} \right)\) A1
hence the line is invariant A1
[3 marks]
hence the eigenvalues are −2 and 3 A1A1
[2 marks]
\(\left( {\begin{array}{*{20}{c}}
1 \\
1
\end{array}} \right)\) and \(\left( {\begin{array}{*{20}{c}}
4 \\
{ - 1}
\end{array}} \right)\) or equivalent A1A1
[2 marks]