How to find a determinant, inverse and eigenvalues of a matrix on the Casio fx-CG50
Enter a matrix, then the determinant, inverse, powers and (on the TI-Nspire) eigenvalues. Exact keystrokes for the fx-CG50 first, then the other calculators, a worked example you can check your answers against, the mistakes that cost marks, and the same thing in Desmos.
When you need it
AI HL: matrix arithmetic, inverses to solve systems, transition matrices and steady states for Markov chains, and eigenvalues/eigenvectors (diagonalisation). AA HL uses matrices only through systems of equations.
Practise it: HL AI matrices · HL AI Markov chains · HL AI eigenvalues.
Keystrokes on your calculator
Pick your calculator: the page remembers it next time.
Matrices on the Casio fx-CG50
Written for OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell). Casio fx-CG50: graphic (GDC).
- MENU 1 (RUN-MATRIX), then F3 (MAT/VCT) to open the matrix list.
- Highlight Mat A, EXE, set m = 2 EXE and n = 2 EXE, type the entries with EXE after each, then EXIT twice.
- Determinant: OPTN F2 (MAT/VCT) F3 (Det), then F1 (Mat) ALPHA X,θ,T (A), EXE.
- Inverse: F1 (Mat) ALPHA X,θ,T (A) SHIFT ) (x⁻¹) EXE. Powers: Mat A ^ 10 for a Markov chain after 10 steps.
- Eigenvalues: there is no eigenvalue command. Find the characteristic equation det(A − λI) = 0 (for a 2 × 2 matrix, λ² − (trace)λ + det = 0) and solve it in EQUATION → POLYNOMIAL.
- Steady state of a Markov chain: raise the transition matrix to a large power (Mat A ^ 50) and check the columns (or rows) stop changing, or solve (T − I)s = 0 with Σs = 1.
- Matrices up to 99 × 99 are possible, but the IB only needs small ones.
Matrices on the TI-84 Plus CE
Written for OS 5.8.x. TI-84 Plus CE: graphic (GDC); also the TI-84 Plus CE Python / CE-T Python Edition.
- 2nd x⁻¹ (MATRIX) → EDIT → 1: [A]. Set 2 × 2 and type the entries.
- 2nd mode (QUIT). Determinant: 2nd x⁻¹ → MATH → 1: det(, then 2nd x⁻¹ → NAMES → 1: [A], ) enter.
- Inverse: [A] x⁻¹ enter (use ►Frac from math for fractions). Powers: [A]^10.
- Eigenvalues: no built-in command; solve λ² − 5λ + 5 = 0 (PlySmlt2 or the Solver).
Matrices on the TI-Nspire CX II
Written for OS 6.x. TI-Nspire CX II: graphic (GDC), non-CAS; the CX II CAS is not allowed in IB, A Level or IGCSE exams.
- Calculator page: templates → the 2 × 2 matrix template, type 2, 1, 1, 3, then ctrl var (sto→) a enter to store it as a.
- det(a) enter (menu → Matrix & Vector → Determinant).
- a^−1 enter for the inverse; a^10 for powers.
- eigVl(a) gives the eigenvalues and eigVc(a) the eigenvectors (unit length) (menu → Matrix & Vector → Eigenvalues / Eigenvectors).
- eigVc gives normalised eigenvectors: scale them to whole numbers if the question expects that.
Worked example
A = (2 1; 1 3). Find det A, A⁻¹ and the eigenvalues of A. Also find det of (1 2 0; 3 −1 4; 2 0 1).
Answers (checked with Python)
det A = 5. A⁻¹ = (0.6 −0.2; −0.2 0.4). Eigenvalues: λ² − 5λ + 5 = 0, so λ = 1.38197 and 3.61803. The 3 × 3 determinant is 9.
Common calculator mistakes
- Entering the matrix by columns instead of rows.
- Multiplying in the wrong order: AB ≠ BA; a transition matrix acts on the state vector from the correct side (check the course's convention).
- Inverse of a singular matrix (det = 0): the calculator gives an error, which usually means a mistake or a system with no unique solution.
- Rounding a transition matrix before raising it to a power.
Do it in Desmos
Not in the exam. Desmos is not an approved calculator for IB exams, so never rely on it for Paper 2. It is excellent for learning, checking a GDC answer and building graphs for your Internal Assessment (IA).
- Matrices are in the separate Desmos Matrix Calculator (desmos.com/matrix): det, inverse and powers, but not eigenvalues.
- In the graphing calculator you can see the eigenvalues as the zeros of the characteristic polynomial: plot y = x² − 5x + 5 (trace 5, det 5) and click the x-intercepts.
The interactive graph loads here when you scroll to it.
Or open desmos.com/calculator and type:
y=x^{2}-5x+5
More calculator skills
Sources
- Casio fx-CG50 Software User's Guide, OS 3.70 (PDF) : Casio fx-CG50, OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell).
- TI-84 Plus CE guidebook (Texas Instruments) : TI-84 Plus CE, OS 5.8.x.
- TI-Nspire CX II guidebooks (Texas Instruments) : TI-Nspire CX II, OS 6.x.
- Desmos Help Center for the Desmos functions.
Menu names are taken from each manufacturer's manual for the OS versions shown. Menus can move slightly between OS updates: if a screen looks different on yours, the manual for your version is the reference, and please tell us so we can update this page. Worked-example answers were recomputed independently in Python.