AHL 1.14—Introduction to matrices Questionbank

Practice AHL 1.14—Introduction to matrices 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 2

Let $M=\left(\begin{array}{cc}a & 2 \\ 2 & -1\end{array}\right)$ , where $a \in \mathbf{Z}$

Find $M^{2}$ in terms of $a$

[2]

If $M^{2}$ is equal to $\left(\begin{array}{cc}5 & -4 \\ -4 & 5\end{array}\right)$ , find the value of $a$ .

[2]

Using this value of $a$ , find $M^{-1}$ and hence solve the system of equations : $\left\{\begin{aligned}-x+2 y & =-3 \\ 2 x-y & =3\end{aligned}\right.$

[5]

Question 2

HLPaper 2

The function $f$ is given by $f(x)=m x^{3}+n x^{2}+p x+q$ , where $m, n, p, q$ are integers. The graph of $f$ passes through the points $(0,0)$ and $(3,18)$

Write down the value of $q$

[1]

Show that $27 m+9 n+2 p=18$

[2]

The graph of $f$ also passes through the points $(1,0)$ and $(-1,-10)$ Write down the other two linear equations in $m, n$ and $p$

[2]

Write down these three equations as a matrix equation.

[2]

Solve this matrix equation.

[2]

The function $f$ can also be written as $f(x)=x(x-1)(r x-s)$ , where $r$ and $s$ are integers. Find $r$ and $s$ .

[3]

Question 3

HLPaper 1

Let $A=\left(\begin{array}{ccc}1 & x & -1 \\ 3 & 1 & 4\end{array}\right)$ and $B=\left(\begin{array}{l}3 \\ x \\ 2\end{array}\right)$

Find $A B$

[2]

The matrix $C=\binom{20}{28}$ and $2 A B=C$ . Find the value of $x$ .

[2]

Question 4

HLPaper 1

Let $\quad A=\left(\begin{array}{lll}3 & 1 & 2 \\ 9 & 5 & 8 \\ 7 & 4 & 6\end{array}\right)$

Write down the value of $\operatorname{det} A$ .

[2]

Write down the inverse of $A$ .

[1]

By investigating the determinant of $A^{n}$ , for several values of $n$ , write down a formula for $\operatorname{det} A^{n}$ in terms of $n$ .

[2]

Question 5

HLPaper 1

Let $A=\left(\begin{array}{ccc}1 & -1 & 3 \\ 2 & 1 & 1 \\ 0 & 2 & -2\end{array}\right)$

Write down $A^{-1}$

[1]

The matrix $B$ satisfies the equation $\left(I-\frac{1}{2} B\right)^{-1}=A$ , where $I$ is the $3 \times 3$ identity matrix.

Show that $B=-2\left(A^{-1}-I\right)$ and hence find $B$

[3]

Write down det $B$ and hence, explain why $B^{-1}$ exists.

[2]

Let $B X=C$ , where

X=\left(\begin{array}{l} x \\ y \\ z \end{array}\right)_{\text {and }} \quad C=\left(\begin{array}{c} 2 \\ -1 \\ 1 \end{array}\right)

Find $X$

[1]

Write down a system of equations whose solution is represented by $X$ .

[3]

Question 6

HLPaper 1

Find the values of the real number $k$ for which the determinant of the matrix is equal to zero.

$\left(\begin{array}{cc}k-4 & -3 \\ -2 & k+1\end{array}\right)$

[4]

Question 7

HLPaper 1

Let $C=\left(\begin{array}{cc}-2 & 4 \\ 1 & 7\end{array}\right)$ and $D=\left(\begin{array}{cc}5 & 2 \\ -1 & a\end{array}\right)$

Find $C D$

[2]

Find $D^{-1}$

[2]

Find $D^{2}$

[2]

Question 8

HLPaper 1

Let $A=\left(\begin{array}{ll}0 & 2 \\ 2 & 0\end{array}\right)$

Find $A^{-1}$

[1]

Find $A^{2}$

[2]

Let $B=\left(\begin{array}{ll}p & 2 \\ 0 & q\end{array}\right)$ Given that $2 A+B=\left(\begin{array}{ll}2 & 6 \\ 4 & 3\end{array}\right)$ , find the value of $p$ and of $q$ .

[3]

Hence, find $A^{-1} B$

[2]

Let $X$ be a $2 \times 2$ matrix such that $A X=B$ . Find $X$ .

[1]

Question 9

HLPaper 2

Let $f(x)=a x^{2}+b x+c$ , where $a, b$ and $c$ are rational numbers.

The point $P(-4,3)$ lies on the curve of $f$ . Show $16 a-4 b+c=3$

[2]

The points $Q(6,3)$ and $R(-2,-1)$ also lie on the curve of $f$ . Write down two other linear equations in $a, b$ and $c$ .

[2]

These three equations may be written as a matrix equation in the form $A X=B$ where $\quad X=\left(\begin{array}{l}a \\ b \\ c\end{array}\right)$

[1]

Write down the matrices $A$ and $B$ .

[2]

Write down $A^{-1}$

[2]

Hence or otherwise, find $f(x)$ .

[1]

Write $f(x)$ in the form $f(x)=a(x-h)^{2}+k$ , where $a, h$ and $k$ are rational numbers.

[1]

Question 10

HLPaper 2

Let $W=\left(\begin{array}{lll}1 & 3 & 2 \\ 2 & 0 & 1 \\ 0 & 1 & 3\end{array}\right)$ and $P=\left(\begin{array}{l}2 \\ 3 \\ 1\end{array}\right)$ and $Q=\left(\begin{array}{l}26 \\ 12 \\ 10\end{array}\right)$

Find $W P$

[1]

Given that $2 W P+S=Q$ , find $S$ .

[2]