CAIE A-Level Further Math 2.2.4 Eigenvalues & eigenvectors

Practise finding real distinct eigenvalues and eigenvectors of 2 by 2 and 3 by 3 matrices from characteristic equations.

Exam points

form det(A-λI)=0 rather than checking only the given eigenvalues
expand and factorise a cubic characteristic equation to find all eigenvalues
find eigenvectors from the rows of A-λI and use them in a diagonal matrix form

[Maximum number: 2]

It is given that a is a positive constant.

Show that the eigenvalues of A are a,-1 and -4.

$\left|\begin{array}{ccc}a-\lambda 2 a+5 a+1 0 -4-\lambda 0 3 -1-\lambda\end{array}\right|=0$

M1

Equates determinant to zero.

$(a-\lambda)(-4-\lambda)(-1-\lambda)=0 \Rightarrow \lambda=a,-1,-4$

A1

AG

2

PUBLISHED

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2.2.4—Eigenvalues and eigenvectors question 1