| Exam Board | Edexcel |
|---|---|
| Module | FP3 (Further Pure Mathematics 3) |
| Marks | 12 |
| Paper | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Matrix properties verification |
| Difficulty | Standard +0.8 This is a Further Maths FP3 question on orthogonal matrices requiring systematic matrix multiplication and solving simultaneous equations. While the concept that MM^T = kI defines an orthogonal matrix (scaled) is standard, students must carefully compute 9 entries, extract multiple equations, and solve a system involving 4 unknowns. The multi-part structure with 12 marks total and the constraint-solving aspect places this moderately above average difficulty. |
| Spec | 4.03a Matrix language: terminology and notation4.03j Determinant 3x3: calculation |
The matrix $\mathbf{M}$ is given by
$$\mathbf{M} = \begin{pmatrix} 1 & 4 & -1 \\ 3 & 0 & p \\ a & b & c \end{pmatrix},$$
where $p$, $a$, $b$ and $c$ are constants and $a > 0$.
Given that $\mathbf{M}\mathbf{M}^T = k\mathbf{I}$ for some constant $k$, find
\begin{enumerate}[label=(\alph*)]
\item the value of $p$, [2]
\item the value of $k$, [2]
\item the values of $a$, $b$ and $c$, [6]
\item $|\det \mathbf{M}|$. [2]
\end{enumerate}
\hfill \mbox{\textit{Edexcel FP3 Q27 [12]}}