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AHL 1.15—Eigenvalues and eigenvectors - IB Questionbank | RevisionDojo

Practice AHL 1.15—Eigenvalues and eigenvectors with authentic IB Mathematics Applications & Interpretation (AI) exam questions for both SL and HL students. This question bank mirrors Paper 1, 2, 3 structure, covering key topics like core principles, advanced applications, and practical problem-solving. Get instant solutions, detailed explanations, and build exam confidence with questions in the style of IB examiners.

Paper

Level

Question 1

HLPaper 1

Let $M=\left(\begin{array}{cc}2 & -1 \\ -3 & 4\end{array}\right)$ and $O=\left(\begin{array}{ll}0 & 0 \\ 0 & 0\end{array}\right)$

Given that $M^{2}-6 M+k I=O$ , find $k \quad$

[6]

Find the characteristic polynomial $\operatorname{det}(A-\lambda I)$

[2]

Write down the eigenvalues of $M$

[3]

Find the corresponding eigenvalues.

[4]

Question 2

HLPaper 1

Let $M=\left(\begin{array}{ll}1 & 1 \\ 1 & 1\end{array}\right)$

Find the eigenvalues of matrix $M$ .

[3]

Find the corresponding eigenvectors.

[4]

The matrix $M$ can be expressed in the form $M=P D P^{-1}$ , where $D$ is a diagonal matrix. Write down the matrices $D$ and $P$ .

[2]

Write down an expression for $D$ in terms of $P$ and $M$ .

[1]

Question 3

HLPaper 1

Consider the matrix $M$ defined as $M=\left(\begin{array}{cc}-1 & k \\ 3 & -1\end{array}\right)$ , where $k$ is a constant. The eigenvalues of $M$ are 2 and -4 .

Find the value of $k$

[3]

Find the corresponding eigenvectors.

[6]

Question 4

HLPaper 2

Let $M=\left(\begin{array}{ll}a & b \\ c & d\end{array}\right)$ be a matrix with real-valued elements $a, b, c, d$ which are such that $a+c=1$ and $b+d=1$

Show that the eigenvalues of $M$ are 1 and $a+d-1$

[4]

Now if $M=I$ , find the eigenvalues and eigenvectors of $M$ .

[4]

Question 5

HLPaper 1

Find a relationship between $a$ and $b$ if the matrices $M = \left( {\begin{array}{*{20}{c}} 1&a \\ 2&3 \end{array}} \right)$ and $N = \left( {\begin{array}{*{20}{c}} 1&b \\ 2&3 \end{array}} \right)$ commute under matrix multiplication.

[4]

Find the value of $a$ if the determinant of matrix $M$ is−1.

[2]

Write down ${M^{ - 1}}$ for thisvalue of $a$ .

[1]

Question 6

HLPaper 1

Find the values of $a$ and $b$ given that the matrix $A = \left( {\begin{array}{*{20}{c}} a&{ - 4}&{ - 6} \\ { - 8}&5&7 \\ { - 5}&3&4 \end{array}} \right)$ is the inverse of the matrix $B = \left( {\begin{array}{*{20}{c}} 1&2&{ - 2} \\ 3&b&1 \\ { - 1}&1&{ - 3} \end{array}} \right)$ .

[2]

For the values of $a$ and $b$ found in part (a), solve the system of linear equations

$\begin{array}{*{20}{c}} {x + 2y - 2z = 5} \\ {3x + by + z = 0} \\ { - x + y - 3z = a - 1.} \end{array}$

[2]

Question 7

HLPaper 3

This question will investigate the solution to a coupled system of differential equations when there is only one eigenvalue.

It is desired to solve the coupled system of differential equations

\dot{x} = 3x + y

\dot{y} = -x + y

The general solution to the coupled system of differential equations is hence given by

\begin{pmatrix} x \\ y \end{pmatrix} = A\begin{pmatrix} 1 \\ -1 \end{pmatrix}e^{2t} + B\begin{pmatrix} t \\ -t + 1 \end{pmatrix}e^{2t}

As $t \to \infty$ the trajectory approaches an asymptote.

State the direction of the trajectory, including the quadrant it is in as it approaches this asymptote.

[1]

Show that the matrix

\left( \begin{array}{cc} 3 & 1 \\ -1 & 1 \end{array} \right)

has (sadly) only one eigenvalue. Find this eigenvalue and an associated eigenvector.

[7]

Verify that

\left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} t \\ -t + 1 \end{array} \right)e^{2t}

is also a solution.

[5]

Find the values of $x$ and $y$ when $t = 2$ .

[2]

Find the equation of this asymptote.

[3]

If initially at $t = 0$ , $x = 20$ , $y = 10$ , find the particular solution.

[3]

Hence, verify that

\left( \begin{array}{c} x \\ y \end{array} \right) = \left( \begin{array}{c} 1 \\ -1 \end{array} \right)e^{2t}

is a solution to the above system.

[5]

Question 8

HLPaper 1

Matrices A, B and C are defined as

A= $\left( {\begin{array}{*{20}{c}} 1&5&1 \\ 3&{ - 1}&3 \\ { - 9}&3&7 \end{array}} \right)$ ,B= $\left( {\begin{array}{*{20}{c}} 1&2&{ - 1} \\ 3&{ - 1}&0 \\ 0&3&1 \end{array}} \right)$ ,C= $\left( {\begin{array}{*{20}{c}} 8 \\ 0 \\ { - 4} \end{array}} \right)$ .

Given that AB = $\left( {\begin{array}{*{20}{c}} a&0&0 \\ 0&a&0 \\ 0&0&a \end{array}} \right)$ , find $a$ .

[1]

Hence, or otherwise, find A^–1.

[2]

Find the matrix X, such that AX = C.

[2]

Question 9

HLPaper 2

Consider the $2 \times 2$ matrix $B$

Given the matrix $B = \begin{pmatrix} -5 & -2 \\ -3 & -4 \end{pmatrix}$ , find the eigenvalues of $B$ .

[4]

Question 10

HLPaper 3

This question will investigate the solution to a coupled system of differential equations and how to transform it to a system that can be solved by the eigenvector method.

It is desired to solve the coupled system of differential equations

$\dot x = x + 2y - 50$

$\dot y = 2x + y - 40$

where $x$ and $y$ represent the population of two types of symbiotic coral and $t$ is time measured in decades.

Find the equilibrium point for this system.

[2]

If initially $x = 100$ and $y = 50$ use Euler’s method with an time increment of 0.1 to find an approximation for the values of $x$ and $y$ when $t = 1$ .

[3]

Extend this method to conjecture the limit of the ratio $\frac{y}{x}$ as $t \to \infty$ .

[2]

Show how using the substitution $X = x - 10{\text{,}}\,\,Y = y - 20$ transforms the system of differential equations into $\begin{array}{*{20}{c}} {\dot X = X + 2Y} \\ {\dot Y = 2X + Y} \end{array}$ .

[3]

Solve this system of equations by the eigenvalue method and hence find the general solution for $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)$ of the original system.

[8]

Find the particular solution to the original system, given the initial conditions of part (b).

[2]

Hence find the exact values of $x$ and $y$ when $t = 1$ , giving the answers to 4 significant figures.

[2]

Use part (f) to find limit of the ratio $\frac{y}{x}$ as $t \to \infty$ .

[2]

With the initial conditions as given in part (b) state if the equilibrium point is stable or unstable.

[1]

10.

If instead the initial conditions were given as $x = 20$ and $y = 10$ , find the particular solution for $\left( {\begin{array}{*{20}{c}} x \\ y \end{array}} \right)$ of the original system, in this case.

[2]

11.

With the initial conditions as given in part (j), determine if the equilibrium point is stable or unstable.

[2]

AHL 1.15—Eigenvalues and eigenvectors Questionbank