| Exam Board | CAIE |
|---|---|
| Module | FP1 (Further Pure Mathematics 1) |
| Year | 2019 |
| Session | June |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | 3x3 Matrices |
| Type | Reconstruct matrix from eigenvalues and eigenvectors |
| Difficulty | Challenging +1.3 This is a structured Further Maths question on diagonalization requiring knowledge of eigenvalues/eigenvectors and matrix reconstruction. Part (i) uses the standard formula A = PDP^(-1), part (ii) applies the inverse, and part (iii) uses the property that A^n applied to an eigenvector gives λ^n times that eigenvector. While requiring multiple techniques, each step follows a well-defined procedure with no novel insight needed, making it moderately above average difficulty for Further Maths students. |
| Spec | 4.03a Matrix language: terminology and notation4.03b Matrix operations: addition, multiplication, scalar4.03n Inverse 2x2 matrix4.03o Inverse 3x3 matrix |
A $3 \times 3$ matrix $\mathbf { A }$ has distinct eigenvalues 2, 1, 3, with corresponding eigenvectors
$$\left( \begin{array} { l }
1 \\
1 \\
0
\end{array} \right) , \quad \left( \begin{array} { r }
- 1 \\
0 \\
b
\end{array} \right) , \quad \left( \begin{array} { r }
0 \\
1 \\
- 1
\end{array} \right)$$
respectively, where $b$ is a positive constant.\\
(i) Find $\mathbf { A }$ in terms of $b$.\\
(ii) Find $\mathbf { A } ^ { - 1 } \left( \begin{array} { r } 0 \\ 2 \\ - 2 \end{array} \right)$.\\
(iii) It is given that
$$\mathbf { A } ^ { n } \left( \begin{array} { l }
1 \\
1 \\
0
\end{array} \right) = \left( \begin{array} { l }
4 \\
4 \\
0
\end{array} \right) \quad \text { and } \quad \mathbf { A } ^ { n } \left( \begin{array} { r }
- 1 \\
0 \\
b
\end{array} \right) = \left( \begin{array} { c }
- 1 \\
0 \\
b ^ { - 1 }
\end{array} \right) .$$
Find the values of $n$ and $b$.\\
\hfill \mbox{\textit{CAIE FP1 2019 Q11 EITHER}}