TI-84 Plus CE lesson 12: Matrices, Markov chains and complex numbers

Matrices and complex numbers are HL topics where the calculator saves the most time: a 3 × 3 system, a Markov chain twenty steps ahead or the modulus and argument of a quotient each take seconds, once you know the MATRIX menu and the a+bi mode.

Lesson 12 of 12
About 55 minutes
TI-84 Plus CE (OS 5.x)
Free, no sign-in

Objective

Enter and use matrices (determinant, inverse, powers, solving AX = B) and work with complex numbers in a+bi mode, including modulus, argument and polar form.

By the end, students can:

enter a matrix with 2nd x⁻¹ (MATRIX), EDIT, and paste it from NAMES
find det( and an inverse, and solve AX = B as [A]⁻¹[B]
raise a transition matrix to a power for a Markov chain, and find the long-run state
set a+bi mode, type i with 2nd ., and use abs(, angle( and ►Polar from math CMPLX

Which courses this lesson suits

IB Maths AI HL
IB Maths AA HL

Matrices and Markov chains are IB AI HL. AA HL students use the complex number steps, and solving a system to check their elimination by hand. The TI-84 Plus CE has no vector menu: check a scalar product with lists, sum(L1*L2), if you need to.

Before you start

Complex results need a+bi on the mode screen; matrices work in any mode. The angle in this lesson is in radians: check the status bar says RADIAN.

The key presses, step by step

Every key is shown as it is printed on the calculator, like 2nd or enter. A second function (printed above a key) is 2nd then that key; a menu item is chosen by its number, or by moving to it and pressing enter.

2nd x⁻¹ opens MATRIX. Move right to the EDIT tab and choose 1: [A].
There are three tabs: NAMES (to paste a matrix into a calculation), MATH (det(, transpose and more) and EDIT (to type one in).
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
Type the size first: 3 enter 3 enter. Then the entries row by row, enter after each: 2, 1, −1 / 1, 3, 2 / 3, −1, 4 (negatives with (−)). Then 2nd mode (QUIT).
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
Determinant: 2nd x⁻¹, MATH tab, 1: det(, then 2nd x⁻¹, NAMES tab, 1: [A], ) enter: 40.
A determinant of 0 means there is no inverse: the system has no unique solution.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
Solve 2x + y − z = 1, x + 3y + 2z = 13, 3x − y + 4z = 13. [A] holds the coefficients; put the right-hand sides in [B] as a 3 × 1 matrix: 2nd x⁻¹ (MATRIX), EDIT tab, 2: [B], type 3 enter 1 enter for the size, then 1, 13, 13 with enter after each, and 2nd mode (QUIT). On the home screen: [A] x⁻¹ [B] enter gives x = 1, y = 2, z = 3.
The inverse key is x⁻¹ itself, pressed after [A]. Paste the names from NAMES: typing the letter A with alpha is a number, not the matrix.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
Complex numbers: mode, move down to the REAL a+bi re^(θi) line, choose a+bi, enter, 2nd mode. i is 2nd .. Type (4 + 3i)/(1 + 2i) enter: 2−i.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
Modulus and argument: math, move right to the CMPLX tab: 5: abs( and 4: angle(. abs(Ans) = 2.236067977 and angle(Ans) = −0.463647609 radians. 7: ►Polar after a number writes it in polar form.
angle( gives the argument between −π and π, in the angle mode the status bar shows.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).

Your turn: check your understanding

Try these straight after the key presses, on your own calculator, before looking at the answers.

Enter [C] = [6 −1; 4 3] (a 2 × 2 matrix) and find det([C]).
In a+bi mode, work out (1 + i)⁴.
Find the modulus and argument of 5 − 12i.

Show the answers

22
−4
abs = 13, angle = −1.18 radians (3 s.f.)

Exam efficiency

Write the matrix equation, e.g. X = A⁻¹B, then the solution: the calculator steps are not working.
Check a solution of a system by putting it back into one equation in your head.
Markov chains: say which convention you used (columns or rows add up to 1), and multiply on the matching side.
For the long-run state, raise the matrix to a large power (50 or 100) and check two powers give the same answer.
Switch back to REAL mode afterwards: in a+bi mode a question with no real solution gives complex answers instead of an error.

Worked exam-style example

Question. Each month, 80% of customers of shop A stay with A and 20% move to B; 70% of B's customers stay and 30% move to A. In month 0, A has 200 customers and B has 800. (a) Find the number with each shop after 5 months. (b) Find the long-run numbers.

Enter the transition matrix [C] = [0.8 0.3; 0.2 0.7] (each column adds up to 1) and the state [D] = [200; 800] as a 2 × 1 matrix.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).
(a) On the home screen: [C]^5[D] enter (^ for the power): A = 587.5, B = 412.5. (b) [C]^50[D] enter: A = 600, B = 400. [C]^100[D] gives the same, so this is the long-run state.
What the screen shows after this step (our drawing of the TI-84 Plus CE screen, not a photo).

What to write down

T = (0.8 0.3; 0.2 0.7), s₀ = (200; 800).

(a) s₅ = T⁵s₀ = (587.5; 412.5).

(b) Tⁿs₀ → (600; 400): 600 with A and 400 with B in the long run.

Practice

Do these on your own calculator. Write the answers to 3 significant figures unless the task says otherwise.

Find the determinant of [4 −2 1; 0 3 5; 2 1 −3].
Find the inverse of [5 2; 3 2]. (Use math 1: ►Frac if you want fractions.)
Solve x + 2y − z = 4, 2x − y + 3z = 9, 3x + y + z = 10.
T = [0.9 0.2; 0.1 0.8] (columns add to 1) and s₀ = [500; 500]. Find s₃ and the long-run state.
Work out (2 + 5i)(3 − i).
Find the modulus and argument of −3 + 4i.

Show the practice answers

−82
[0.5 −0.5; −0.75 1.25]
x = 1.4, y = 2.8, z = 3
s₃ = (609.5; 390.5); long run (667; 333) to 3 s.f.
11+13i
abs = 5, angle = 2.21 radians (3 s.f.)

Common mistakes

Typing A with alpha instead of pasting [A] from NAMES: you get a number, not the matrix.
[B][A]⁻¹ instead of [A]⁻¹[B]: matrix order matters.
Entering a Markov matrix with rows and columns swapped, so the columns do not add up to 1.
ERROR: NONREAL ANSWERS for √(−4) and similar real inputs in REAL mode: switch to a+bi. (Typing i yourself gives a complex result even in REAL mode.)
Typing i with alpha instead of 2nd ..
Giving the argument in degrees when the question wants radians (or the other way round).

1-minute quick check

Answer these without looking back. Most need no calculation: they check that the key presses have stuck.

Which keys open the MATRIX menu?
How do you solve AX = B on the home screen?
Where is i?
Which mode must be on for complex answers?
Where are abs( and angle(?

Show the quick-check answers

2nd x⁻¹
[A] x⁻¹ [B] enter (names pasted from NAMES)
2nd .
a+bi (the mode screen)
math, CMPLX tab

Teaching this lesson (about 55 minutes)

0–5 min	Starter: by hand, det of [3 1; 2 4]. Then check on the calculator.
5–20 min	Steps 1–4: the matrix editor, det(, solving a 3 × 3 system. First "your turn" task.
20–30 min	Worked example: the Markov chain, 5 steps and the long run.
30–38 min	Steps 5–6: a+bi mode, i, abs(, angle(, ►Polar. Last two "your turn" tasks.
38–52 min	Practice 1–6 (AA HL: 3, 5 and 6).
52–55 min	1-minute quick check.

Students need their own TI-84 Plus CE. With a projector, the calculator-skills guides below have a Show on the board view and an animated walkthrough of the key presses.