Matrices — multiplication, inverse, and determinant (IB Math AI HL)

Key ideas

• AB exists only if A has as many columns as B has rows. In general AB ≠ BA, so keep the order.
• 2×2: det A = ad − bc and A⁻¹ = (1/(ad − bc)) × (d −b; −c a). Use the GDC for 3×3 determinants and inverses.
• A system of linear equations is AX = B, so X = A⁻¹B: multiply by A⁻¹ on the left.
• If det A = 0, A has no inverse and the system has either no solution or infinitely many.
• (AB)⁻¹ = B⁻¹A⁻¹. For a transformation matrix, |det A| is the area scale factor.

Worked example

Use an inverse matrix to solve 2x + 3y = 7 and 4x − y = 7.

1. A = (2 3; 4 −1), X = (x; y), B = (7; 7).
2. det A = 2 × (−1) − 3 × 4 = −14.
3. A⁻¹ = (1/−14)(−1 −3; −4 2).
4. X = A⁻¹B = (1/−14)(−7 − 21; −28 + 14) = (1/−14)(−28; −14) = (2; 1).
5. Check: 2(2) + 3(1) = 7 and 4(2) − 1 = 7.

Answer: x = 2, y = 1

Exam-style HL AI questions on this topic

• Matrices, Determinants and Inverses questions — 42 questions with mark schemes, plus free worked examples
• All HL AI Number and Algebra questions by topic

Do it on your GDC

A GDC is allowed on every AI paper. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:

• Matrices on your calculator
• Try it: matrix calculator (free, shows the working)

Maths IA ideas that use matrices

• Animating a logo with transformation matrices
• Will one phone brand take over? Brand loyalty as a Markov chain
• Encrypting with matrices: when can a Hill cipher be undone?
• Projecting a country's age structure with a Leslie matrix
• All HL AI IA ideas

Related in Number and Algebra

• Systems of linear equations and Gaussian elimination
• Complex numbers and Argand diagrams
• Sequences, series, and the sum to infinity

Revise Matrices for the exam

• HL AI revision notes — every unit summarised, with the key results to learn
• HL AI past papers — specimen papers and timed practice papers
• Formulas for Number and Algebra — on the one-page HL AI formula sheet (free PDF), with the ones not in the IB booklet marked
• HL AI formula booklet by topic — every formula in the IB booklet's notation, with a free PDF
• IB Maths prediction papers — full papers in the IB format, with markschemes

TOK link: Do mathematicians have ethical responsibilities for how their work is used?

Stretch yourself: original algebra problems at Senior level (ages 16 to 18), with hints and full solutions: Problem S24 · Problem S88 · Problem S23 · Problem S87. All 22 →