Pre-U Pre-U 9795/1 2011 June — Question 1 4 marks

Exam BoardPre-U
ModulePre-U 9795/1 (Pre-U Further Mathematics Paper 1)
Year2011
SessionJune
Marks4
TopicMatrices
TypeMatrix satisfying given equation
DifficultyStandard +0.8 This question requires finding a parameter k such that A³ = I, which involves matrix multiplication and solving a system of equations. While the computation is manageable, it requires careful algebraic manipulation across multiple matrix products and isn't a standard textbook exercise. The conceptual leap that det(A³) = (det A)³ = det(I) = 1 provides a useful constraint. This is moderately challenging for a Further Maths question but not exceptionally difficult.
Spec4.03h Determinant 2x2: calculation4.03o Inverse 3x3 matrix
Given that the matrix \(\mathbf{A} = \begin{pmatrix} 2 & k \\ 1 & -3 \end{pmatrix}\), where \(k\) is real, is such that \(\mathbf{A}^3 = \mathbf{I}\), find the value of \(k\) and the numerical value of \(\det \mathbf{A}\). [4]
AnswerMarks Guidance
\(A^2 = \begin{pmatrix} k+4 & -k \\ -1 & k+9 \end{pmatrix}\), \(A^3 = \begin{pmatrix} k+8 & k^2+7k \\ k+7 & -4k-27 \end{pmatrix}\)M1 Attempt at \(A^2\) and \(A^3\)
\(A^3 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \Leftrightarrow k=-7\)A1 \(\geq 3\) entries of \(A^3\) checked
All entries of \(A^3\) must be verified if done this wayA1 Otherwise, allow just one key element checked (since "given")
\(\det\begin{pmatrix} 2 & -7 \\ 1 & -3 \end{pmatrix} = -6-k = 1\) ft numerical value consistent with their \(k\)B1
\(k = -7\)A1
[4]
ALTERNATIVE
AnswerMarks
\(\det(A^3) = (\det A)^3\)M1
\(A^3 = I \Rightarrow \det A = 1\)A1
\(\det A = -6-k\)B1
\(k = -7\)A1
[4] 
$A^2 = \begin{pmatrix} k+4 & -k \\ -1 & k+9 \end{pmatrix}$, $A^3 = \begin{pmatrix} k+8 & k^2+7k \\ k+7 & -4k-27 \end{pmatrix}$ | M1 | Attempt at $A^2$ and $A^3$
$A^3 = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \Leftrightarrow k=-7$ | A1 | $\geq 3$ entries of $A^3$ checked
All entries of $A^3$ must be verified if done this way | A1 | Otherwise, allow just one key element checked (since "given")
$\det\begin{pmatrix} 2 & -7 \\ 1 & -3 \end{pmatrix} = -6-k = 1$ ft numerical value consistent with their $k$ | B1 | 
$k = -7$ | A1 |
| [4] |

**ALTERNATIVE**
$\det(A^3) = (\det A)^3$ | M1 |
$A^3 = I \Rightarrow \det A = 1$ | A1 |
$\det A = -6-k$ | B1 |
$k = -7$ | A1 |
| [4] |

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Given that the matrix $\mathbf{A} = \begin{pmatrix} 2 & k \\ 1 & -3 \end{pmatrix}$, where $k$ is real, is such that $\mathbf{A}^3 = \mathbf{I}$, find the value of $k$ and the numerical value of $\det \mathbf{A}$. [4]

\hfill \mbox{\textit{Pre-U Pre-U 9795/1 2011 Q1 [4]}}
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