Practise questions · Salam Academy

SA-FP-EIG-0011 markL1 FoundationNon-calculatorEigenvalues

A

3\times3

matrix has trace

9

and two eigenvalues

2

and

3

. Find the third eigenvalue.

Salam Academy Original Practice · Proposed mark scheme

SA-FP-EIG-Q011 markL1 FoundationNon-calculatorEigenvectors

If

\mathbf v

is an eigenvector of

\mathbf M

with eigenvalue

3

, then

\mathbf M(2\mathbf v)=

Salam Academy Original Practice · Proposed mark scheme

SA-FP-EIG-X041 markL1 FoundationNon-calculatorEigenvalues

A

2\times2

matrix has trace

7

and determinant

10

. Find the sum of its eigenvalues.

Salam Academy Original Practice · Proposed mark scheme

SA-FP-EIG-0022 marksL2 StandardNon-calculatorEigenvectors

\begin{pmatrix}1\\k\end{pmatrix}

is an eigenvector of

\begin{pmatrix}4&1\\2&3\end{pmatrix}

with eigenvalue

2

. Find

k

.

Salam Academy Original Practice · Proposed mark scheme

SA-FP-EIG-G012 marksL2 StandardNon-calculatorEigenvalues

Find the eigenvalues of

\begin{pmatrix}4&1\\2&3\end{pmatrix}

.

Salam Academy Original Practice · Proposed mark scheme

SA-FP-EIG-X012 marksL2 StandardNon-calculatorEigenvalues

Find the eigenvalues of

\begin{pmatrix}4&1\\2&3\end{pmatrix}

.

Salam Academy Original Practice · Proposed mark scheme

SA-FP-EIG-X022 marksL2 StandardNon-calculatorEigenvectors

For the same matrix, an eigenvector for

\lambda=5

is

\begin{pmatrix}1\\k\end{pmatrix}

. Find

k

.

Salam Academy Original Practice · Proposed mark scheme

SA-FP-EIG-X032 marksL2 StandardNon-calculatorEigenvectors

For the same matrix, an eigenvector for

\lambda=2

is

\begin{pmatrix}1\\k\end{pmatrix}

. Find

k

.

Salam Academy Original Practice · Proposed mark scheme

SA-FP-EIG-0033 marksL3 ExamNon-calculatorDiagonalisation

\mathbf M=\mathbf{PDP}^{-1}

with

\mathbf D=\begin{pmatrix}2&0\\0&-1\end{pmatrix}

. Find the trace of

\mathbf M^5

.

Salam Academy Original Practice · Proposed mark scheme

SA-FP-EIG-0043 marksL3 ExamNon-calculatorEigenvalues

Find the eigenvalues of

\begin{pmatrix}2&0&0\\1&3&0\\4&5&-1\end{pmatrix}

.

Salam Academy Original Practice · Proposed mark scheme

SA-FP-EIG-X052 marksL3 ExamNon-calculatorDiagonalisation

\mathbf M=\mathbf{PDP}^{-1}

with

\mathbf D=\begin{pmatrix}2&0\\0&5\end{pmatrix}

. Find

\det(\mathbf M^3)

.

Salam Academy Original Practice · Proposed mark scheme

SA-FP-EIG-0053 marksL4 ChallengingNon-calculatorDiagonalisation

\mathbf M

satisfies

\mathbf M^2=5\mathbf M-6\mathbf I

. Express

\mathbf M^3=p\mathbf M+q\mathbf I

. Find

p

.

Salam Academy Original Practice · Proposed mark scheme

Question bank