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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

Find a relationship between $a$ and $b$ if the matrices $M = \begin{pmatrix} 1 & a \\ 2 & 3 \end{pmatrix}$ and $N = \begin{pmatrix} 1 & b \\ 2 & 3 \end{pmatrix}$ commute under matrix multiplication.

[4]

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

[2]

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

[1]

Question 2

HLPaper 1

Let $M = \begin{pmatrix} 2 & -1 \\ -3 & 4 \end{pmatrix}$ and $O = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ .

Given that $M^{2} - 6M + kI = O$ , find $k$ .

[6]

Find the characteristic polynomial $\det(M - \lambda I)$ .

[2]

Write down the eigenvalues of $M$ .

[3]

Find the eigenvectors corresponding to the eigenvalues found in part 3.

[4]

Question 3

HLPaper 1

Let $M=\begin{pmatrix}1 & 1 \\ 1 & 1\end{pmatrix}$

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 4

HLPaper 1

Find the values of $a$ and $b$ given that the matrix $A = \begin{pmatrix} a & -4 & -6 \\ -8 & 5 & 7 \\ -5 & 3 & 4 \end{pmatrix}$ is the inverse of the matrix $B = \begin{pmatrix} 1 & 2 & -2 \\ 3 & b & 1 \\ -1 & 1 & -3 \end{pmatrix}$ .

[2]

For the values of $a$ and $b$ found in part 1, solve the system of linear equations

\begin{aligned} x + 2y - 2z &= 5 \\ 3x + by + z &= 0 \\ -x + y - 3z &= a - 1 \end{aligned}

[2]

Question 5

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}

Show that the matrix

\begin{pmatrix} 3 & 1 \\ -1 & 1 \end{pmatrix}

has only one eigenvalue. Find this eigenvalue and an associated eigenvector.

[7]

Hence, verify that

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

is a solution to the above system.

[5]

Verify that

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

is also a solution.

[5]

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

[3]

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

[2]

Determine the limiting value of the ratio $\frac{y}{x}$ as $t \to \infty$ and hence state the equation of the line passing through the origin to which the trajectory becomes parallel.

[3]

State the quadrant in which the trajectory lies as $t \to \infty$ and describe its motion relative to the origin.

[1]

Question 6

HLPaper 1

Consider the matrix $M$ defined as $M=\begin{pmatrix}-1 & k \\ 3 & -1\end{pmatrix}$ , 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 7

HLPaper 1

Matrices $A$ , $B$ and $C$ are defined as: $A = \begin{pmatrix} 1 & 5 & 1 \\ 3 & -1 & 3 \\ -9 & 3 & 7 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & 2 & -1 \\ 3 & -1 & 0 \\ 0 & 3 & 1 \end{pmatrix}$ $C = \begin{pmatrix} 8 \\ 0 \\ -4 \end{pmatrix}$

Given that $AB = \begin{pmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{pmatrix}$ , find $a$ .

[1]

Hence, or otherwise, find $A^{-1}$ .

[2]

Find the matrix $X$ , such that $AX = C$ .

[2]

Question 8

HLPaper 2

Let $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ be a matrix with real-valued elements $a, b, c, d$ 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 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 a 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 $X = x - 10, \, Y = y - 20$ transforms the system of differential equations into $\begin{aligned} \dot{X} &= X + 2Y \\ \dot{Y} &= 2X + Y \end{aligned}$

[3]

Solve this system of equations by the eigenvalue method and hence find the general solution for $\begin{pmatrix} x \\ y \end{pmatrix}$ of the original system.

[8]

Find the particular solution to the original system, given the initial conditions of part 2.

[2]

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

[2]

Use part 6 to find limit of the ratio $\frac{y}{x}$ as $t \to \infty$ .

[2]

With the initial conditions as given in part 2, determine if the populations converge to the equilibrium values.

[1]

10.

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

[2]

11.

With the initial conditions as given in part 10, determine if the populations converge to the equilibrium values.

[2]

Paper

Level

Question 1

HLPaper 1

Find a relationship between $a$ and $b$ if the matrices $M = \begin{pmatrix} 1 & a \\ 2 & 3 \end{pmatrix}$ and $N = \begin{pmatrix} 1 & b \\ 2 & 3 \end{pmatrix}$ commute under matrix multiplication.

[4]

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

[2]

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

[1]

Question 2

HLPaper 1

Let $M = \begin{pmatrix} 2 & -1 \\ -3 & 4 \end{pmatrix}$ and $O = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ .

Given that $M^{2} - 6M + kI = O$ , find $k$ .

[6]

Find the characteristic polynomial $\det(M - \lambda I)$ .

[2]

Write down the eigenvalues of $M$ .

[3]

Find the eigenvectors corresponding to the eigenvalues found in part 3.

[4]

Question 3

HLPaper 1

Let $M=\begin{pmatrix}1 & 1 \\ 1 & 1\end{pmatrix}$

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 4

HLPaper 1

[2]

For the values of $a$ and $b$ found in part 1, solve the system of linear equations

\begin{aligned} x + 2y - 2z &= 5 \\ 3x + by + z &= 0 \\ -x + y - 3z &= a - 1 \end{aligned}

[2]

Question 5

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}

Show that the matrix

\begin{pmatrix} 3 & 1 \\ -1 & 1 \end{pmatrix}

has only one eigenvalue. Find this eigenvalue and an associated eigenvector.

[7]

Hence, verify that

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

is a solution to the above system.

[5]

Verify that

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

is also a solution.

[5]

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

[3]

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

[2]

Determine the limiting value of the ratio $\frac{y}{x}$ as $t \to \infty$ and hence state the equation of the line passing through the origin to which the trajectory becomes parallel.

[3]

State the quadrant in which the trajectory lies as $t \to \infty$ and describe its motion relative to the origin.

[1]

Question 6

HLPaper 1

Consider the matrix $M$ defined as $M=\begin{pmatrix}-1 & k \\ 3 & -1\end{pmatrix}$ , 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 7

HLPaper 1

Given that $AB = \begin{pmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{pmatrix}$ , find $a$ .

[1]

Hence, or otherwise, find $A^{-1}$ .

[2]

Find the matrix $X$ , such that $AX = C$ .

[2]

Question 8

HLPaper 2

Let $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ be a matrix with real-valued elements $a, b, c, d$ 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 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

Find the equilibrium point for this system.

[2]

If initially $x = 100$ and $y = 50$ , use Euler's method with a 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 $X = x - 10, \, Y = y - 20$ transforms the system of differential equations into $\begin{aligned} \dot{X} &= X + 2Y \\ \dot{Y} &= 2X + Y \end{aligned}$

[3]

Solve this system of equations by the eigenvalue method and hence find the general solution for $\begin{pmatrix} x \\ y \end{pmatrix}$ of the original system.

[8]

Find the particular solution to the original system, given the initial conditions of part 2.

[2]

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

[2]

Use part 6 to find limit of the ratio $\frac{y}{x}$ as $t \to \infty$ .

[2]

With the initial conditions as given in part 2, determine if the populations converge to the equilibrium values.

[1]

10.

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

[2]

11.

With the initial conditions as given in part 10, determine if the populations converge to the equilibrium values.

[2]

Paper

Level

Question 1

HLPaper 1

Find a relationship between $a$ and $b$ if the matrices $M = \begin{pmatrix} 1 & a \\ 2 & 3 \end{pmatrix}$ and $N = \begin{pmatrix} 1 & b \\ 2 & 3 \end{pmatrix}$ commute under matrix multiplication.

[4]

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

[2]

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

[1]

Question 2

HLPaper 1

Let $M = \begin{pmatrix} 2 & -1 \\ -3 & 4 \end{pmatrix}$ and $O = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$ .

Given that $M^{2} - 6M + kI = O$ , find $k$ .

[6]

Find the characteristic polynomial $\det(M - \lambda I)$ .

[2]

Write down the eigenvalues of $M$ .

[3]

Find the eigenvectors corresponding to the eigenvalues found in part 3.

[4]

Question 3

HLPaper 1

Let $M=\begin{pmatrix}1 & 1 \\ 1 & 1\end{pmatrix}$

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 4

HLPaper 1

[2]

For the values of $a$ and $b$ found in part 1, solve the system of linear equations

\begin{aligned} x + 2y - 2z &= 5 \\ 3x + by + z &= 0 \\ -x + y - 3z &= a - 1 \end{aligned}

[2]

Question 5

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}

Show that the matrix

\begin{pmatrix} 3 & 1 \\ -1 & 1 \end{pmatrix}

has only one eigenvalue. Find this eigenvalue and an associated eigenvector.

[7]

Hence, verify that

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

is a solution to the above system.

[5]

Verify that

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

is also a solution.

[5]

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

[3]

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

[2]

Determine the limiting value of the ratio $\frac{y}{x}$ as $t \to \infty$ and hence state the equation of the line passing through the origin to which the trajectory becomes parallel.

[3]

State the quadrant in which the trajectory lies as $t \to \infty$ and describe its motion relative to the origin.

[1]

Question 6

HLPaper 1

Consider the matrix $M$ defined as $M=\begin{pmatrix}-1 & k \\ 3 & -1\end{pmatrix}$ , 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 7

HLPaper 1

Given that $AB = \begin{pmatrix} a & 0 & 0 \\ 0 & a & 0 \\ 0 & 0 & a \end{pmatrix}$ , find $a$ .

[1]

Hence, or otherwise, find $A^{-1}$ .

[2]

Find the matrix $X$ , such that $AX = C$ .

[2]

Question 8

HLPaper 2

Let $M = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ be a matrix with real-valued elements $a, b, c, d$ 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 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

Find the equilibrium point for this system.

[2]

If initially $x = 100$ and $y = 50$ , use Euler's method with a 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 $X = x - 10, \, Y = y - 20$ transforms the system of differential equations into $\begin{aligned} \dot{X} &= X + 2Y \\ \dot{Y} &= 2X + Y \end{aligned}$

[3]

Solve this system of equations by the eigenvalue method and hence find the general solution for $\begin{pmatrix} x \\ y \end{pmatrix}$ of the original system.

[8]

Find the particular solution to the original system, given the initial conditions of part 2.

[2]

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

[2]

Use part 6 to find limit of the ratio $\frac{y}{x}$ as $t \to \infty$ .

[2]

With the initial conditions as given in part 2, determine if the populations converge to the equilibrium values.

[1]

10.

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

[2]

11.

With the initial conditions as given in part 10, determine if the populations converge to the equilibrium values.

[2]

AHL 1.15—Eigenvalues and eigenvectors Questionbank