CAIE A-Level Further Math A2 2.2.4 Eigenvalues and Eigenvectors Questions

Practise finding eigenvalues from characteristic equations and matrix data, checking roots and interpreting the resulting matrix properties with Further Mathematics Paper 2 questions.

Exam points

Find eigenvalues from characteristic equations and matrix data.

[Maximum number: 2]

The matrix A is given by

\mathbf{A}=\left(\begin{array}{ccc} a & -6 a & 2 a+2 \\ 0 & 1-a & 0 \\ 0 & 2-a & -1 \end{array}\right)

where a is a constant with $a \neq 0$ and $a \neq 1$ .

Show that the eigenvalues of A are a, 1-a and -1.

$\left|\begin{array}{ccc}a-\lambda -6 a 2 a+2 0 1-a-\lambda 0 2-a -1-\lambda\end{array}\right|=0$
Equates determinant to zero or working to find
characteristic equation.

$(a-\lambda)(1-a-\lambda)(-1-\lambda)=0 \Rightarrow \lambda=a, 1-a,-1$
AG. Factorisation must be clear.

2

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CAIE A-Level Further Math A2 2.2.4 Eigenvalues and Eigenvectors Questions question 1