IB Maths AI HL · Unit 1: Number and Algebra
IB Maths AI HL Matrices, Determinants and Inverses Questions
Exam-style IB Maths AI HL matrices, determinants and inverses questions with worked solutions. Start with the 3 fully worked examples below — each is solved step by step the way an IB examiner expects — then try the practice questions and check your working against the mark scheme.
- 21 questions
- Paper 1: 21
- 5 easy
- 5 medium
- 10 hard
- 1 starter
- 3 worked examples
Practise Matrices, Determinants and Inverses questions →
AI HL formula booklet
What's examined in AI HL matrices, determinants and inverses
The question bank covers these matrices, determinants and inverses question types (number of questions in brackets):
- Matrix Algebra Basics (7)
- Determinants and Singularity (7)
- Inverses and Equations (7)
Matrices, Determinants and Inverses worked examples
Worked example 1: Executing Matrix Multiplication · easy
Consider the matrices $A = \begin{pmatrix} 4 & -2 \\ 1 & 5 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 3 \\ -1 & 2 \end{pmatrix}$. Calculate exactly the matrix product $AB$.
1. Verify the dimensions: both are $2 \times 2$ matrices, so they can be multiplied to produce a $2 \times 2$ result.
2. Calculate the top-left element (Row 1 $\times$ Col 1): $(4)(0) + (-2)(-1) = 0 + 2 = 2$.
3. Calculate the top-right element (Row 1 $\times$ Col 2): $(4)(3) + (-2)(2) = 12 - 4 = 8$.
4. Calculate the bottom-left element (Row 2 $\times$ Col 1): $(1)(0) + (5)(-1) = 0 - 5 = -5$.
5. Calculate the bottom-right element (Row 2 $\times$ Col 2): $(1)(3) + (5)(2) = 3 + 10 = 13$. The final matrix is $\begin{pmatrix} 2 & 8 \\ -5 & 13 \end{pmatrix}$.
Examiner tip: Matrix multiplication operates by multiplying the rows of the first matrix by the columns of the second matrix; it is generally not commutative ($AB \neq BA$).
Worked example 2: Using Matrix Inverses to Solve Systems · medium
A system of equations is written in matrix form as $MX = C$, where $M = \begin{pmatrix} 2 & 1 & -1 \\ 1 & 2 & -1 \\ 3 & -1 & 2 \end{pmatrix}$, $X = \begin{pmatrix} x \\ y \\ z \end{pmatrix}$, and $C = \begin{pmatrix} 1 \\ 2 \\ 7 \end{pmatrix}$. Using your Graphic Display Calculator, find the exact values of $x$, $y$, and $z$.
1. Rearrange the matrix equation to solve for $X$ by pre-multiplying both sides by the inverse of $M$: $X = M^{-1}C$.
2. Enter matrix $M$ and matrix $C$ into the matrix workspace of your GDC.
3. Compute the inverse matrix $M^{-1}$ using the $x^{-1}$ function on your calculator.
4. Multiply $M^{-1}$ by $C$ directly on your GDC to prevent intermediate rounding errors.
5. Verify the exact solutions by direct substitution: $2(1) + 2 - 3 = 1$; $1 + 2(2) - 3 = 2$; $3(1) - 2 + 2(3) = 7$. All three equations balance, so $x = 1$, $y = 2$, and $z = 3$.
Examiner tip: When isolating $X$ in the equation $MX = C$, you must strictly pre-multiply by the inverse to get $X = M^{-1}C$. Post-multiplying ($C M^{-1}$) is dimensionally impossible here.
Worked example 3: Identifying Singular Matrices · hard
Find the exact values of $k$ for which the matrix $A = \begin{pmatrix} k & 4 \\ 3 & k-1 \end{pmatrix}$ is singular.
1. Recall that a matrix is singular if and only if its determinant is exactly zero.
2. Set up the determinant formula for a $2 \times 2$ matrix ($\det(A) = ad - bc$): $k(k-1) - (4)(3) = 0$.
3. Expand the brackets: $k^2 - k - 12 = 0$.
4. Factorise the resulting quadratic equation: $(k - 4)(k + 3) = 0$.
5. Solve to find the exact values that make the matrix singular: $k = 4$ and $k = -3$.
Examiner tip: A singular matrix cannot be inverted because dividing by its determinant (which is zero) is mathematically undefined. This means the associated system of equations has no unique solution.
Try these IB Maths AI HL matrices, determinants and inverses questions
Three questions from the bank, easiest first. Mark schemes and AI marking of your written working are in the practice area.
Question 1 · easy · 4 marks · Paper 1
A matrix \(M\) is defined as \(M = \begin{pmatrix} 6 & 2 \\ 5 & 3 \end{pmatrix}\).
(a) Calculate the determinant of \(M\), denoted as \(\det(M)\) or \(|M|\). [2 marks]
(b) Hence, write down the exact inverse matrix, \(M^{-1}\). [2 marks]
Question 2 · medium · 5 marks · Paper 1
A company sells two types of bundles:
Bundle A contains 3 phones and 2 cases, and costs €1850.
Bundle B contains 4 phones and 5 cases, and costs €2700.
Let \(p\) be the price of one phone and \(c\) be the price of one case.
(a) Write this information as a system of linear equations in the form \(AX = B\), where \(X = \begin{pmatrix} p \\ c \end{pmatrix}\). [2 marks]
(b) Find \(A^{-1}\) and use it to calculate the exact price of one phone and one case. [3 marks]
Question 3 · hard · 8 marks · Paper 1
Consider the matrix $A = \begin{pmatrix} x & 2 \\ -1 & y \end{pmatrix}$, where $x, y \in \mathbb{R}$.
Given that $A^2 = I$, where $I$ is the $2 \times 2$ identity matrix:
(a) Find the possible values for $x$ and $y$. [5 marks]
(b) Using your answers from part (a), determine the value of $\det(2A)$. [3 marks]
All 21 matrices, determinants and inverses questions with mark schemes →
FAQ
How many IB Maths AI HL matrices, determinants and inverses questions are there?
There are 21 exam-style matrices, determinants and inverses questions in the AI HL question bank (Paper 1: 21), graded 5 easy, 5 medium, 10 hard, 1 starter. Every question has a full IB-style mark scheme (M, A and R marks).
Is matrices, determinants and inverses on Paper 1 or Paper 2?
In the question bank these questions are set as Paper 1 questions.
Where can I get the mark schemes?
Open the AI HL Unit 1 practice page: every question has a step-by-step IB-style mark scheme, and you can photograph your working for instant AI marking. The worked examples on this page are free.
← All IB Maths AI HL topics