Matrices (AI HL)
A matrix is a rectangular array of numbers used to organise and manipulate data - from transforming shapes to solving systems of equations. This topic covers the basic algebra of matrices (adding, scaling and multiplying them), the determinant and inverse of a 2×2 matrix, and how to use an inverse matrix to solve a system of linear equations written as \(A\mathbf{x}=\mathbf{b}\).
What the syllabus says
This topic maps onto one point in the official IB Applications & Interpretation syllabus, HL-only.
| Code | Syllabus content |
|---|---|
| AHL1.14 | Definition of a matrix: the terms element, row, column and order for \(m\times n\) matrices. Algebra of matrices: equality, addition, subtraction, multiplication by a scalar. Multiplication of matrices, and its properties (associativity, distributivity, non-commutativity). Identity and zero matrices. Determinants and inverses of \(n\times n\) matrices with technology, and by hand for \(2\times2\) matrices. Writing a system of linear equations as \(A\mathbf{x}=\mathbf{b}\) and solving using the inverse matrix; in examinations \(A\) is always invertible, except when solving for eigenvectors. |
AHL1.14 is HL-only content, not examinable at AI SL.
Key terms
Five words worth knowing cold before you touch the formulas below - each with a worked example showing exactly what it means.
What is the order of a matrix?
The order of a matrix is its number of rows by its number of columns, written \(m\times n\). It tells you the matrix's shape and, along with another matrix's order, whether the two can be added or multiplied together.
e.g. \(\begin{pmatrix}1&2&3\\4&5&6\end{pmatrix}\) has order \(2\times3\): 2 rows, 3 columns.
What is scalar multiplication of a matrix?
Scalar multiplication multiplies every single element of a matrix by the same number. It's the matrix equivalent of multiplying a vector by a constant, and the result keeps the same order as the original.
e.g. \(A=\begin{pmatrix}1&2\\3&4\end{pmatrix}\): \(3A=\begin{pmatrix}3&6\\9&12\end{pmatrix}\).
What is matrix multiplication?
Matrix multiplication combines the rows of the first matrix with the columns of the second: each entry of the product is the sum of the products of corresponding row and column elements. It's only defined when the number of columns in the first matrix equals the number of rows in the second.
e.g. \(\begin{pmatrix}1&2\\3&4\end{pmatrix}\begin{pmatrix}2&0\\1&3\end{pmatrix}=\begin{pmatrix}4&6\\10&12\end{pmatrix}\).
What is the determinant of a 2×2 matrix?
The determinant is a single number computed from a matrix's entries, \(\det A = ad-bc\) for \(A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\). If the determinant is zero, the matrix is singular and has no inverse.
e.g. \(A=\begin{pmatrix}5&2\\3&4\end{pmatrix}\): \(\det A = 5(4)-2(3)=14\).
What is the inverse of a matrix?
The inverse \(A^{-1}\) is the matrix that "undoes" \(A\), so that \(AA^{-1}=A^{-1}A=I\), the identity matrix. It's used to solve a matrix equation \(A\mathbf{x}=\mathbf{b}\) by computing \(\mathbf{x}=A^{-1}\mathbf{b}\).
e.g. \(A=\begin{pmatrix}2&1\\1&1\end{pmatrix}\): \(A^{-1}=\begin{pmatrix}1&-1\\-1&2\end{pmatrix}\), since \(AA^{-1}=I\).
Key formulas
Two hand-calculable formulas cover 2×2 matrices; everything larger is handled by your GDC. The tables below summarise all of it at a glance.
Formula reference
The 2×2 determinant and inverse formulas are on the official formula booklet. Basic matrix algebra (addition, scalar multiplication) follows directly from the definitions and isn't listed as a separate formula.
| Formula | Used for | Booklet? |
|---|---|---|
| \(\det A = ad-bc\) | Determinant of \(A=\begin{pmatrix}a&b\\c&d\end{pmatrix}\) | ✓ Yes |
| \(A^{-1}=\dfrac{1}{\det A}\begin{pmatrix}d&-b\\-c&a\end{pmatrix}\) | Inverse of a 2×2 matrix | ✓ Yes |
| \(A\mathbf{x}=\mathbf{b}\ \Rightarrow\ \mathbf{x}=A^{-1}\mathbf{b}\) | Solving a linear system | Not listed separately - apply the inverse formula above |
| Addition, scalar multiplication | Combining matrices element-by-element | Not in the formula booklet - definition |
By hand vs by technology
The syllabus draws a clear line: 2×2 matrices are examinable by hand, anything larger relies on your GDC.
| Feature | 2×2 matrices | Larger matrices (3×3 and up) |
|---|---|---|
| Determinant | By hand: \(ad-bc\) | By technology (GDC matrix mode) |
| Inverse | By hand, using the formula above | By technology only |
| Solving \(A\mathbf{x}=\mathbf{b}\) | Either method works | By technology: \([A]^{-1}[B]\) |
| In examinations | \(A\) is always invertible | \(A\) is always invertible |
Matrix algebra
Matrices add and scale element-by-element like ordinary arithmetic, but multiplication behaves very differently from multiplying numbers.
Addition & subtraction
Add or subtract corresponding elements. Matrices must have the same order - you can't add a \(2\times2\) matrix to a \(2\times3\) one.
Not in the formula booklet - definitionScalar multiplication
Multiply every element by the same number. Combines naturally with addition, e.g. \(2A+B\).
Not in the formula booklet - definitionMatrix multiplication
Only defined when the columns of the first matrix match the rows of the second. Associative and distributive, but \(AB\neq BA\) in general - order matters.
Not in the formula booklet - definitionDeterminants, inverses & solving systems
These three ideas connect directly: the determinant tells you whether an inverse exists, and the inverse is exactly what solving \(A\mathbf{x}=\mathbf{b}\) needs.
Singular matrices
\[\det A = 0 \Rightarrow A^{-1}\text{ does not exist}\]
A singular matrix has no inverse, and the corresponding system of equations has no unique solution.
Not a formula - consequence of the inverse formulaIdentity and zero matrices
The identity matrix \(I\) leaves any matrix unchanged under multiplication (\(AI=IA=A\)); the zero matrix \(O\) has every entry \(0\).
Not in the formula booklet - definitionWriting a system as \(A\mathbf{x}=\mathbf{b}\)
Collect the coefficients of a linear system into a matrix \(A\), the unknowns into a column vector \(\mathbf{x}\), and the constants into \(\mathbf{b}\) - then solve with \(\mathbf{x}=A^{-1}\mathbf{b}\).
Not a formula - modelling techniqueWorked examples
Two full exam-style questions, marked exactly like the real thing. Try each one yourself before checking the worked solution.
Let \(A=\begin{pmatrix}2&-1\\0&3\end{pmatrix}\) and \(B=\begin{pmatrix}4&1\\5&-2\end{pmatrix}\):
(a) Find \(2A+B\).
(b) Find \(A-2B\).
Worked solution
(a) \(2A=\begin{pmatrix}4&-2\\0&6\end{pmatrix}.\) M1
Adding: \(2A+B=\begin{pmatrix}8&-1\\5&4\end{pmatrix}.\) A1
(b) \(\begin{pmatrix}-6&-3\\-10&7\end{pmatrix}.\) M1 A1
Use the inverse matrix to solve \(\begin{cases}4x+3y=18\\2x+y=8\end{cases}\).
Worked solution
\(A=\begin{pmatrix}4&3\\2&1\end{pmatrix},\ \mathbf b=\begin{pmatrix}18\\8\end{pmatrix}.\) M1
\(\det A=-2;\ \mathbf x=A^{-1}\mathbf b\) M1
\(=\begin{pmatrix}3\\2\end{pmatrix},\) so \(x=3,\ y\) A1
\(=2.\) A1
Common mistakes
The four slip-ups that account for most of the marks lost on this topic - worth reading before you start practising, not just after you get one wrong.
- Assuming matrix multiplication is commutative. \(AB\) and \(BA\) are generally different matrices (or one might not even be defined) - never swap the order without checking.
- Multiplying matrices with incompatible orders. \(AB\) only exists when the number of columns in \(A\) equals the number of rows in \(B\) - check the orders before attempting the multiplication.
- Forgetting a zero determinant means no inverse. If \(\det A=0\), the matrix is singular, \(A^{-1}\) doesn't exist, and \(A\mathbf{x}=\mathbf{b}\) has no unique solution - though the IB guarantees \(A\) is invertible in examinations.
- Sign errors in the 2×2 inverse formula. Swap \(a\) and \(d\), but only negate \(b\) and \(c\) - a common slip is negating all four entries, or forgetting to divide by the determinant.
Using your GDC
Every step below is a real button sequence, not a vague "use your calculator" hint - covering the TI-84 Plus, TI-Nspire, and Casio fx-9860/fx-CG50. Pick your model to filter down to just the steps that apply to you.
The two pre-matched calculator skills for this topic - entering scientific notation and the numeric equation solver - aren't the real workhorse here. This topic's actual GDC skill is your calculator's dedicated matrix mode: you store matrices in matrix memory, then add, multiply and compute determinants and inverses directly on them, which is exactly how the two worked examples above are checked (see each solution's calculator note). Determinants and inverses of anything larger than 2×2 are found only using this mode - it's not examined by hand.
The exact keystrokes for entering and naming a matrix differ by model - TI-84 uses the MATRIX menu, TI-Nspire uses the matrix template on the keyboard, and Casio uses the Matrix/Run-Matrix app - so see the full GDC guide for the precise steps on your model rather than a generic summary here.
See the full GDC guide for more calculator models and topics.
Ready to practise properly?
Matrices questions, marked instantly like the real exam.
Quick answers
The questions students on this topic ask most often.
Can I multiply any two matrices together?
No - matrix multiplication \(AB\) is only defined when the number of columns in \(A\) equals the number of rows in \(B\). If \(A\) is \(2\times3\) and \(B\) is \(3\times2\), \(AB\) is defined and gives a \(2\times2\) matrix, but \(BA\) would need checking separately, since matrix multiplication is not commutative.
Why isn't matrix multiplication commutative?
Because \(AB\) and \(BA\) combine the rows and columns of \(A\) and \(B\) in a different order, so they generally give different results (or one might not even be defined). This is different from ordinary number multiplication, where order never matters.
When does a matrix not have an inverse?
A matrix has no inverse when its determinant is zero - it's called singular. For a \(2\times2\) matrix with \(\det A = ad-bc=0\), the inverse formula would require dividing by zero, and the associated system of equations has no unique solution.
Can I use my GDC for matrices?
Yes, and for anything bigger than \(2\times2\) you're expected to. Every GDC has a matrix mode for storing matrices and computing sums, products, determinants and inverses directly - you're only required to find determinants and inverses by hand for \(2\times2\) matrices. See the GDC guide for model-specific instructions.
Sub-topics
Matrices broken down into its individual skills, each with its own focused page.
Related topics
More Number & Algebra topics from the same AI HL syllabus unit, in case you want to keep going.