| Date | May 2018 | Marks available | 3 | Reference code | 18M.1.hl.TZ0.2 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Show that | Question number | 2 | Adapted from | N/A |
Question
Let A2 = 2A + I where A is a 2 × 2 matrix.
Show that A4 = 12A + 5I.
Let B = \(\left[ {\begin{array}{*{20}{c}}
4&2 \\
1&{ - 3}
\end{array}} \right]\).
Given that B2 – B – 4I = \(\left[ {\begin{array}{*{20}{c}}
k&0 \\
0&k
\end{array}} \right]\), find the value of \(k\).
Markscheme
METHOD 1
A4 = 4A2 + 4AI + I2 or equivalent M1A1
= 4(2A + I) + 4A + I A1
= 8A + 4I + 4A + I
= 12A + 5I AG
[3 marks]
METHOD 2
A3 = A(2A + I) = 2A2 + AI = 2(2A + I) + A(= 5A + 2I) M1A1
A4 = A(5A + 2I) A1
= 5A2 + 2A = 5(2A + I) + 2A
= 12A + 5I AG
[3 marks]
B2 = \(\left[ {\begin{array}{*{20}{c}}
{18}&2 \\
1&{11}
\end{array}} \right]\) (A1)
\(\left[ {\begin{array}{*{20}{c}}
{18}&2 \\
1&{11}
\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}
4&2 \\
1&{ - 3}
\end{array}} \right] - \left[ {\begin{array}{*{20}{c}}
4&0 \\
0&4
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
{10}&0 \\
0&{10}
\end{array}} \right]\) (A1)
\( \Rightarrow k = 10\) A1
[3 marks]
