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AHL 1.14—Introduction to matrices - IB Questionbank | RevisionDojo

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 2

Consider the following matrices: $A = \begin{pmatrix} 2 & -1 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 5 & 2 \\ -1 & 3 \end{pmatrix}$

Calculate $A + B$ and $B + A$ to verify that matrix addition is commutative.

[3]

Find $2A$ and verify that $(2A) + B = A + A + B$ .

[4]

Question 5

HLPaper 2

Consider matrices $P = \begin{pmatrix} 2 & 1 \\ 3 & 4 \end{pmatrix}$ and $Q = \begin{pmatrix} 5 & 2 \\ 1 & 3 \end{pmatrix}$

Determine whether these matrices are commutative.

[3]

Question 6

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 7

HLPaper 1

Let A = $\left( {\begin{array}{*{20}{c}} 1&2 \\ 3&{ - 1} \end{array}} \right)$ andB= $\left( {\begin{array}{*{20}{c}} 3&0 \\ { - 2}&1 \end{array}} \right)$ .

Find A + B.

[2]

Find −3A.

[2]

Find AB.

[3]

Question 8

HLPaper 2

The matrix M is given by M $= \left[ {\begin{array}{*{20}{c}} 1&2&2 \\ 3&1&1 \\ 2&3&1 \end{array}} \right]$ .

Given that M³ can be written as a quadratic expression in M in the formaM² + bM+ cI , determine the values of the constants a, b and c.

[7]

Show that M⁴=19M²+40M+30I.

[2]

Using mathematical induction, prove that Mⁿ can be written as a quadratic expression in M for all positive integers n≥ 3.

[6]

Find a quadratic expression in M for the inverse matrix M^–1.

[2]

Question 9

HLPaper 1

The matrices A, B, C and X are all non-singular 3 × 3 matrices.

Given that A^–1XB = C, express X in terms of the other matrices.

Question 10

HLPaper 1

Consider the matrix A = $\left( {\begin{array}{*{20}{c}} 0&2 \\ a&{ - 1} \end{array}} \right)$ .

Find the matrix A².

[2]

If det A²= 16, determine the possible values of $a$ .

[3]

AHL 1.14—Introduction to matrices Questionbank