Diagonalisation and the Cayley-Hamilton theoremEdexcel A-Level Further Maths: Subtopic test

1. 1
   The matrix $\mathbf{A}=\begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}$ has eigenvalues $2$ and $5$, with eigenvectors $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 1 \end{pmatrix}$ respectively. Let $\mathbf{P}=\begin{pmatrix} 1 & 1 \\ -2 & 1 \end{pmatrix}$.

   (a)

   Which is $\mathbf{P}^{-1}\mathbf{A}\mathbf{P}$?

   [1 mark]

   • A$\begin{pmatrix} 5 & 0 \\ 0 & 2 \end{pmatrix}$
   • B$\begin{pmatrix} 4 & 0 \\ 0 & 3 \end{pmatrix}$
   • C$\begin{pmatrix} 2 & 1 \\ 0 & 5 \end{pmatrix}$
   • D$\begin{pmatrix} 2 & 0 \\ 0 & 5 \end{pmatrix}$

   (b)

   Find $\det\mathbf{P}$.

   [1 mark]

   • A$-1$
   • B$1$
   • C$3$
   • D$-3$

   (c)

   Write down a matrix $\mathbf{Q}$ such that $\mathbf{Q}^{-1}\mathbf{A}\mathbf{Q}=\begin{pmatrix} 5 & 0 \\ 0 & 2 \end{pmatrix}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The symmetric matrix $\mathbf{S}=\begin{pmatrix} 5 & 2 \\ 2 & 2 \end{pmatrix}$.

   (a)

   Find the eigenvalues of $\mathbf{S}$.

   [1 mark]

   • A$1$ and $6$
   • B$2$ and $5$
   • C$-1$ and $-6$
   • D$0$ and $7$

   (b)

   Which is a normalised eigenvector of $\mathbf{S}$ corresponding to the eigenvalue $6$?

   [1 mark]

   • A$\begin{pmatrix} 2 \\ 1 \end{pmatrix}$
   • B$\frac{1}{\sqrt5}\begin{pmatrix} 2 \\ 1 \end{pmatrix}$
   • C$\frac{1}{\sqrt3}\begin{pmatrix} 2 \\ 1 \end{pmatrix}$
   • D$\frac{1}{\sqrt5}\begin{pmatrix} 1 \\ -2 \end{pmatrix}$

   (c)

   Find an orthogonal matrix $\mathbf{Q}$ and a diagonal matrix $\mathbf{D}$ such that $\mathbf{Q}^{\mathrm{T}}\mathbf{S}\mathbf{Q}=\mathbf{D}$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The matrix $\mathbf{C}=\begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$.

   (a)

   Find the characteristic equation of $\mathbf{C}$ and hence show that $\mathbf{C}^2=6\mathbf{C}-5\mathbf{I}$.

   [3 marks]

   (b)

   Hence find $\mathbf{C}^{-1}$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The matrix $\mathbf{M}=\begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$.

   (a)

   Find the eigenvalues and corresponding eigenvectors of $\mathbf{M}$, and hence write down a matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that $\mathbf{P}^{-1}\mathbf{M}\mathbf{P}=\mathbf{D}$.

   [6 marks]

   (b)

   Using your answer to part (a), show that $\mathbf{M}^n=\begin{pmatrix} 2^n & 3^n-2^n \\ 0 & 3^n \end{pmatrix}$ for positive integers $n$.

   [6 marks]

   Total for question 4: 12 marks

End of questions