Diagonalisation of symmetric matricesEdexcel International A Level Further Maths: Subtopic test

1. 1
   The symmetric matrix $\mathbf{S}=\begin{pmatrix} 3 & 1 \\ 1 & 3 \end{pmatrix}$.

   (a)

   What are the eigenvalues of $\mathbf{S}$?

   [1 mark]

   • A$0$ and $6$
   • B$1$ and $5$
   • C$2$ and $4$
   • D$3$ and $3$

   (b)

   Which orthogonal matrix $\mathbf{P}$ gives $\mathbf{P}^T\mathbf{S}\mathbf{P}=\begin{pmatrix} 2 & 0 \\ 0 & 4 \end{pmatrix}$?

   [1 mark]

   • A$\frac{1}{\sqrt2}\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}$
   • B$\begin{pmatrix} 1 & 1 \\ -1 & 1 \end{pmatrix}$
   • C$\frac{1}{\sqrt2}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}$
   • D$\frac{1}{\sqrt2}\begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix}$

   (c)

   Show that eigenvectors of $\mathbf{S}$ corresponding to its two different eigenvalues are perpendicular.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The symmetric matrix $\mathbf{T}=\begin{pmatrix} 2 & 0 & 0 \\ 0 & 5 & 2 \\ 0 & 2 & 2 \end{pmatrix}$ can be reduced to diagonal form by an orthogonal matrix $\mathbf{P}$, so that $\mathbf{P}^T\mathbf{T}\mathbf{P}$ is diagonal.

   (a)

   Which statement about $\mathbf{P}$ must be true?

   [1 mark]

   • A$\det\mathbf{P}=1$
   • B$\mathbf{P}^T=\mathbf{P}^{-1}$
   • C$\mathbf{P}$ is symmetric
   • DAll entries of $\mathbf{P}$ are integers

   (b)

   The columns of $\mathbf{P}$ are normalised eigenvectors of $\mathbf{T}$, written in order of increasing eigenvalue. What is $\mathbf{P}^T\mathbf{T}\mathbf{P}$?

   [1 mark]

   • A$\begin{pmatrix} 2 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 6 \end{pmatrix}$
   • B$\begin{pmatrix} 6 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 1 \end{pmatrix}$
   • C$\begin{pmatrix} 2 & 0 & 0 \\ 0 & 5 & 0 \\ 0 & 0 & 2 \end{pmatrix}$
   • D$\begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 6 \end{pmatrix}$

   (c)

   Find a normalised eigenvector of $\mathbf{T}$ corresponding to the eigenvalue $6$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The symmetric matrix $\mathbf{S}=\begin{pmatrix} 1 & 2 \\ 2 & -2 \end{pmatrix}$.

   (a)

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

   [3 marks]

   (b)

   Find an orthogonal matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that $\mathbf{P}^T\mathbf{S}\mathbf{P}=\mathbf{D}$, with the eigenvalue $2$ first.

   [4 marks]

   Total for question 3: 7 marks

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

   (a)

   (i) Find the eigenvalues of $\mathbf{A}$.
   (ii) Verify that the sum of the eigenvalues equals the trace of $\mathbf{A}$ and that their product equals $\det\mathbf{A}$.

   [6 marks]

   (b)

   Find an orthogonal matrix $\mathbf{P}$ and a diagonal matrix $\mathbf{D}$ such that $\mathbf{P}^T\mathbf{A}\mathbf{P}=\mathbf{D}$, with the eigenvalues in increasing order on the diagonal of $\mathbf{D}$.

   [6 marks]

   Total for question 4: 12 marks

End of questions