HL AI · Number and Algebra

Systems of linear equations and Gaussian elimination (IB Math AI HL)

Updated October 2026

Quick answer: Write the system as AX = B, where A is the matrix of coefficients. If det A ≠ 0 there is exactly one solution, X = A⁻¹B, which the GDC finds directly. If det A = 0 there is no unique solution: either no solution or infinitely many.

HL AI expects you to solve 3×3 systems by matrix inversion OR row reduction. Know when a system has unique, infinite, or no solutions from the determinant.

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Key ideas

Each equation is one row of A, with the coefficients of x, y and z in the same order; B is the column of right-hand sides.
det A ≠ 0: one solution, X = A⁻¹B. Multiply A⁻¹ on the left of B; the order matters.
det A = 0: the equations are either inconsistent (no solution) or dependent (infinitely many); the GDC's matrix inverse will fail.
Write the matrix equation in your answer before giving the solution, so the method is clear.
Check the solution by substituting into every original equation, not just one.

Worked example

Solve the system x + y + z = 6, 2x − y + z = 3, x + 2y − z = 2.

Matrix form: A = (1 1 1; 2 −1 1; 1 2 −1), X = (x, y, z), B = (6, 3, 2).
det A = 7 ≠ 0, so there is exactly one solution.
X = A⁻¹B. On the GDC: x = 1, y = 2, z = 3.
Check: 1 + 2 + 3 = 6; 2 − 2 + 3 = 3; 1 + 4 − 3 = 2.

Answer: x = 1, y = 2, z = 3.

Exam-style HL AI questions

HL AI Number and Algebra questions by topic — worked examples and practice questions for every topic in this unit

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:

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Formulas for Number and Algebra — on the one-page HL AI formula sheet (free PDF), with the ones not in the IB booklet marked
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FAQ

How do you know if a system of equations has a unique solution?

Write it as AX = B and find the determinant of A. If det A is not zero, there is exactly one solution, X = A⁻¹B. If det A is zero, there is either no solution or infinitely many, and the inverse does not exist.

Can I use my calculator for systems of linear equations and Gaussian elimination?

Yes: a GDC is allowed on every HL AI paper, so practise the calculator method as well as the working. The Solving equations and Matrices guides give the keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50.

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