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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 $M=\begin{pmatrix}a & 2 \\ 2 & -1\end{pmatrix}$ , where $a \in \mathbb{Z}$

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

[2]

If $M^{2}$ is equal to $\begin{pmatrix}5 & -4 \\ -4 & 5\end{pmatrix}$ , 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

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 3

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 4

HLPaper 2

The function $f$ is given by $f(x) = mx^3 + nx^2 + px + q$ , 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 $27m + 9n + 3p = 18$ .

[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 $f(x) = x(x - 1)(rx - s)$ , where $r$ and $s$ are integers. Find $r$ and $s$ .

[3]

Question 5

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 6

HLPaper 1

Let $A = \begin{pmatrix} 1 & 2 \\ 3 & -1 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 0 \\ -2 & 1 \end{pmatrix}$.

Find $A + B$.

[2]

Find $-3A$.

[2]

Find $AB$.

[3]

Question 7

HLPaper 2

The matrix $\boldsymbol{M}$ is given by $\boldsymbol{M} = \begin{bmatrix} 1 & 2 & 2 \\ 3 & 1 & 1 \\ 2 & 3 & 1 \end{bmatrix}$ .

Given that $\boldsymbol{M}^3$ can be written as a quadratic expression in $\boldsymbol{M}$ in the form $a\boldsymbol{M}^2 + b\boldsymbol{M} + c\boldsymbol{I}$ , determine the values of the constants $a$ , $b$ and $c$ .

[7]

Show that $\boldsymbol{M}^4 = 19\boldsymbol{M}^2 + 40\boldsymbol{M} + 30\boldsymbol{I}$ .

[2]

Using mathematical induction, prove that $\boldsymbol{M}^n$ can be written as a quadratic expression in $\boldsymbol{M}$ for all positive integers $n \geq 3$ .

[6]

Find a quadratic expression in $\boldsymbol{M}$ for the inverse matrix $\boldsymbol{M}^{-1}$ .

[2]

Question 8

HLPaper 1

The matrices $\mathbf{A}$ , $\mathbf{B}$ , $\mathbf{C}$ and $\mathbf{X}$ are all non-singular $3 \times 3$ matrices.

Given that $\mathbf{A}^{-1}\mathbf{X}\mathbf{B} = \mathbf{C}$ , express $\mathbf{X}$ in terms of the other matrices.

[3]

Question 9

HLPaper 1

Let $A = \begin{pmatrix} 1 & x & -1 \\ 3 & 1 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 3 \\ x \\ 2 \end{pmatrix}$ .

Find $AB$ .

[2]

The matrix $C = \begin{pmatrix} 20 \\ 28 \end{pmatrix}$ and $2AB = C$ . Find the value of $x$ .

[2]

Question 10

HLPaper 1

Consider the matrix $A = \begin{pmatrix} 0 & 2 \\ a & -1 \end{pmatrix}$ .

Find the matrix $A^2$ .

[2]

If $\det A^2 = 16$ , determine the possible values of $a$ .

[3]

Paper

Level

Question 1

HLPaper 2

Let $M=\begin{pmatrix}a & 2 \\ 2 & -1\end{pmatrix}$ , where $a \in \mathbb{Z}$

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

[2]

If $M^{2}$ is equal to $\begin{pmatrix}5 & -4 \\ -4 & 5\end{pmatrix}$ , 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

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 3

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 4

HLPaper 2

The function $f$ is given by $f(x) = mx^3 + nx^2 + px + q$ , 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 $27m + 9n + 3p = 18$ .

[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 $f(x) = x(x - 1)(rx - s)$ , where $r$ and $s$ are integers. Find $r$ and $s$ .

[3]

Question 5

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 6

HLPaper 1

Let $A = \begin{pmatrix} 1 & 2 \\ 3 & -1 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 0 \\ -2 & 1 \end{pmatrix}$.

Find $A + B$.

[2]

Find $-3A$.

[2]

Find $AB$.

[3]

Question 7

HLPaper 2

The matrix $\boldsymbol{M}$ is given by $\boldsymbol{M} = \begin{bmatrix} 1 & 2 & 2 \\ 3 & 1 & 1 \\ 2 & 3 & 1 \end{bmatrix}$ .

[7]

Show that $\boldsymbol{M}^4 = 19\boldsymbol{M}^2 + 40\boldsymbol{M} + 30\boldsymbol{I}$ .

[2]

Using mathematical induction, prove that $\boldsymbol{M}^n$ can be written as a quadratic expression in $\boldsymbol{M}$ for all positive integers $n \geq 3$ .

[6]

Find a quadratic expression in $\boldsymbol{M}$ for the inverse matrix $\boldsymbol{M}^{-1}$ .

[2]

Question 8

HLPaper 1

The matrices $\mathbf{A}$ , $\mathbf{B}$ , $\mathbf{C}$ and $\mathbf{X}$ are all non-singular $3 \times 3$ matrices.

Given that $\mathbf{A}^{-1}\mathbf{X}\mathbf{B} = \mathbf{C}$ , express $\mathbf{X}$ in terms of the other matrices.

[3]

Question 9

HLPaper 1

Let $A = \begin{pmatrix} 1 & x & -1 \\ 3 & 1 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 3 \\ x \\ 2 \end{pmatrix}$ .

Find $AB$ .

[2]

The matrix $C = \begin{pmatrix} 20 \\ 28 \end{pmatrix}$ and $2AB = C$ . Find the value of $x$ .

[2]

Question 10

HLPaper 1

Consider the matrix $A = \begin{pmatrix} 0 & 2 \\ a & -1 \end{pmatrix}$ .

Find the matrix $A^2$ .

[2]

If $\det A^2 = 16$ , determine the possible values of $a$ .

[3]

Paper

Level

Question 1

HLPaper 2

Let $M=\begin{pmatrix}a & 2 \\ 2 & -1\end{pmatrix}$ , where $a \in \mathbb{Z}$

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

[2]

If $M^{2}$ is equal to $\begin{pmatrix}5 & -4 \\ -4 & 5\end{pmatrix}$ , 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

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 3

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 4

HLPaper 2

The function $f$ is given by $f(x) = mx^3 + nx^2 + px + q$ , 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 $27m + 9n + 3p = 18$ .

[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 $f(x) = x(x - 1)(rx - s)$ , where $r$ and $s$ are integers. Find $r$ and $s$ .

[3]

Question 5

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 6

HLPaper 1

Let $A = \begin{pmatrix} 1 & 2 \\ 3 & -1 \end{pmatrix}$ and $B = \begin{pmatrix} 3 & 0 \\ -2 & 1 \end{pmatrix}$.

Find $A + B$.

[2]

Find $-3A$.

[2]

Find $AB$.

[3]

Question 7

HLPaper 2

The matrix $\boldsymbol{M}$ is given by $\boldsymbol{M} = \begin{bmatrix} 1 & 2 & 2 \\ 3 & 1 & 1 \\ 2 & 3 & 1 \end{bmatrix}$ .

[7]

Show that $\boldsymbol{M}^4 = 19\boldsymbol{M}^2 + 40\boldsymbol{M} + 30\boldsymbol{I}$ .

[2]

Using mathematical induction, prove that $\boldsymbol{M}^n$ can be written as a quadratic expression in $\boldsymbol{M}$ for all positive integers $n \geq 3$ .

[6]

Find a quadratic expression in $\boldsymbol{M}$ for the inverse matrix $\boldsymbol{M}^{-1}$ .

[2]

Question 8

HLPaper 1

The matrices $\mathbf{A}$ , $\mathbf{B}$ , $\mathbf{C}$ and $\mathbf{X}$ are all non-singular $3 \times 3$ matrices.

Given that $\mathbf{A}^{-1}\mathbf{X}\mathbf{B} = \mathbf{C}$ , express $\mathbf{X}$ in terms of the other matrices.

[3]

Question 9

HLPaper 1

Let $A = \begin{pmatrix} 1 & x & -1 \\ 3 & 1 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 3 \\ x \\ 2 \end{pmatrix}$ .

Find $AB$ .

[2]

The matrix $C = \begin{pmatrix} 20 \\ 28 \end{pmatrix}$ and $2AB = C$ . Find the value of $x$ .

[2]

Question 10

HLPaper 1

Consider the matrix $A = \begin{pmatrix} 0 & 2 \\ a & -1 \end{pmatrix}$ .

Find the matrix $A^2$ .

[2]

If $\det A^2 = 16$ , determine the possible values of $a$ .

[3]

AHL 1.14—Introduction to matrices Questionbank