Diagonalisation of matricesAQA A-Level Further Maths: Subtopic test

1. 1
   The matrix $M=\begin{pmatrix} 3 & 1 \\ 2 & 4 \end{pmatrix}$ has eigenvalues $2$ and $5$, with corresponding eigenvectors $\begin{pmatrix} 1 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$.

   (a)

   Which of these pairs of matrices $U$ and $D$ satisfies $M=UDU^{-1}$?

   [1 mark]

   • A$U=\begin{pmatrix} 1 & 1 \\ -1 & 2 \end{pmatrix}$, $D=\begin{pmatrix} 5 & 0 \\ 0 & 2 \end{pmatrix}$
   • B$U=\begin{pmatrix} 1 & -1 \\ 1 & 2 \end{pmatrix}$, $D=\begin{pmatrix} 2 & 0 \\ 0 & 5 \end{pmatrix}$
   • C$U=\begin{pmatrix} 1 & 1 \\ -1 & 2 \end{pmatrix}$, $D=\begin{pmatrix} 2 & 0 \\ 0 & 5 \end{pmatrix}$
   • D$U=\begin{pmatrix} 1 & 1 \\ 2 & -1 \end{pmatrix}$, $D=\begin{pmatrix} 2 & 0 \\ 0 & 5 \end{pmatrix}$

   (b)

   What is the eigenvalue of $M^3$ corresponding to the eigenvector $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$?

   [1 mark]

   • A$125$
   • B$15$
   • C$25$
   • D$8$

   (c)

   Taking $U=\begin{pmatrix} 1 & 1 \\ -1 & 2 \end{pmatrix}$, find $U^{-1}$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The matrix $A=\begin{pmatrix} 2 & 1 \\ 1 & 2 \end{pmatrix}$ can be written as $A=UDU^{-1}$, where $U=\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$ and $D$ is a diagonal matrix.

   (a)

   Which of these is $D$?

   [1 mark]

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

   (b)

   What is the entry in the first row and first column of $A^4$?

   [1 mark]

   • A$81$
   • B$41$
   • C$40$
   • D$82$

   (c)

   Explain why $A^4$ has the same eigenvectors as $A$, and state the eigenvalues of $A^4$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The matrix $B=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 3 & 1 \\ 0 & 1 & 3 \end{pmatrix}$ has eigenvalues $1$, $2$ and $4$, with corresponding eigenvectors $\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$, $\begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}$.

   (a)

   Write down a matrix $U$ and a diagonal matrix $D$ such that $B=UDU^{-1}$.

   [3 marks]

   (b)

   Hence find $B^5$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The matrix $M=\begin{pmatrix} -2 & 3 \\ -6 & 7 \end{pmatrix}$.

   (a)

   Find a matrix $U$ and a diagonal matrix $D$ such that $M=UDU^{-1}$, and find $U^{-1}$.

   [6 marks]

   (b)

   Using your matrices from (a), show that $M^n=\begin{pmatrix} 2-4^n & 4^n-1 \\ 2-2\times4^n & 2\times4^n-1 \end{pmatrix}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions