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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 the eigenvalues and corresponding eigenvectors of the following matrix

$A=\left(\begin{array}{cc}2 & \frac{-7}{3} \\ \frac{13}{6} & \frac{-5}{2}\end{array}\right)$ .

[6]

Question 6

HLPaper 2

Let $A=\left(\begin{array}{cc}2 & 1 \\ -3 & 2\end{array}\right)$ and $B=\left(\begin{array}{ll}2 & 1 \\ 3 & 2\end{array}\right)$

Show that $A$ has no real eigenvalues.

[2]

Find the eigenvalues of $B$ in the form $a \pm \sqrt{b}$ , where $a, b \in \mathbf{Z}$

[2]

Find the corresponding eigenvectors of $B$

[4]

Question 7

HLPaper 2

Consider the matrix $M=\left(\begin{array}{cc}3 & -2 \\ p & 1\end{array}\right)$ , where $p$ is a constant and $\binom{1}{2}$ is an eigenvector of $M$ .

Find the value of $p$ .

[4]

Question 8

HLPaper 2

Let $M=\left(\begin{array}{ll}0.8 & 0.1 \\ 0.2 & 0.9\end{array}\right)$

Show $M^{n}=\frac{1}{3}\left(\begin{array}{ll}1+2 \times 0.7^{n} & 1-0.7^{n} \\ 2-2 \times 0.7^{n} & 2+0.7^{n}\end{array}\right)$

[6]

Deduce the result of $M^{n}$ as $n$ tends to infinity.

[2]

Question 9

HLPaper 2

Let $A=\left(\begin{array}{ll}0.1 & 0.4 \\ 0.9 & 0.6\end{array}\right)$

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

[2]

Hence find the eigenvalues of matrix $A$ .

[2]

Find the corresponding eigenvectors.

[6]

Question 10

HLPaper 2

Find the eigenvalues and corresponding eigenvectors of the following matrix

$B=\left(\begin{array}{cc}0.1 & 0.5 \\ -0.02 & 0.3\end{array}\right)$ .

[6]

AHL 1.15—Eigenvalues and eigenvectors Questionbank