| Exam Board | Edexcel |
|---|---|
| Module | F3 (Further Pure Mathematics 3) |
| Year | 2021 |
| Session | January |
| Marks | 6 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Determinant calculation and singularity |
| Difficulty | Standard +0.3 This is a standard Further Maths question requiring determinant calculation of a 3×3 matrix with a parameter, solving a cubic equation for singularity, and finding the inverse using the adjugate method. While it involves more computation than Core modules, it's routine application of well-practiced techniques without requiring novel insight or complex problem-solving. |
| Spec | 4.03l Singular/non-singular matrices4.03o Inverse 3x3 matrix |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| \( | \mathbf{A} | = 2(4-2k) - k(4-k) + 2(4-2) = 0\) |
| \(k^2 - 8k + 12 = 0 \Rightarrow k = 2, 6\) | A1 | Correct values |
| Answer | Marks | Guidance |
|---|---|---|
| Answer/Working | Mark | Guidance |
| Matrix of minors with correct sign pattern \(\begin{pmatrix}+ & - & +\\ - & + & -\\ + & - & +\end{pmatrix}\) applied | M1 | Applies correct method to reach at least a matrix of cofactors |
| \(\begin{pmatrix}4-2k & 4-2k & k^2-4\\ k-4 & 2 & 4-2k\\ 2 & k-4 & 4-2k\end{pmatrix}\) | dM1 A1 | dM1: Attempts adjoint by transposing. A1: Correct adjoint |
| \(\mathbf{A}^{-1} = \dfrac{1}{k^2-8k+12}\begin{pmatrix}4-2k & 4-2k & k^2-4\\ k-4 & 2 & 4-2k\\ 2 & k-4 & 4-2k\end{pmatrix}\) | A1ft | Fully correct inverse or follow through their incorrect determinant from part (a) where determinant is a function of \(k\) |
# Question 3(a):
| Answer/Working | Mark | Guidance |
|---|---|---|
| $|\mathbf{A}| = 2(4-2k) - k(4-k) + 2(4-2) = 0$ | M1 | Attempts det $\mathbf{A} = 0$ and solves 3TQ to obtain 2 values for $k$. Allow errors only when collecting terms |
| $k^2 - 8k + 12 = 0 \Rightarrow k = 2, 6$ | A1 | Correct values |
# Question 3(b):
| Answer/Working | Mark | Guidance |
|---|---|---|
| Matrix of minors with correct sign pattern $\begin{pmatrix}+ & - & +\\ - & + & -\\ + & - & +\end{pmatrix}$ applied | M1 | Applies correct method to reach at least a matrix of cofactors |
| $\begin{pmatrix}4-2k & 4-2k & k^2-4\\ k-4 & 2 & 4-2k\\ 2 & k-4 & 4-2k\end{pmatrix}$ | dM1 A1 | dM1: Attempts adjoint by transposing. A1: Correct adjoint |
| $\mathbf{A}^{-1} = \dfrac{1}{k^2-8k+12}\begin{pmatrix}4-2k & 4-2k & k^2-4\\ k-4 & 2 & 4-2k\\ 2 & k-4 & 4-2k\end{pmatrix}$ | A1ft | Fully correct inverse or follow through their incorrect determinant from part (a) where determinant is a function of $k$ |3.
$$\mathbf { A } = \left( \begin{array} { l l l }
2 & k & 2 \\
2 & 2 & k \\
1 & 2 & 2
\end{array} \right) \quad \text { where } k \text { is a constant }$$
\begin{enumerate}[label=(\alph*)]
\item Determine the values of $k$ for which $\mathbf { A }$ is singular.
Given that $\mathbf { A }$ is non-singular,
\item find $\mathbf { A } ^ { - 1 }$, giving your answer in terms of $k$.\\
3.
\end{enumerate}
\hfill \mbox{\textit{Edexcel F3 2021 Q3 [6]}}