HL · Matrices 1 · What a matrix is: order, adding and scaling

Module 12 of 47Lesson 1 of 8 · HL · Matrices 1 · What a matrix is: order, adding and scaling

Matrices & eigenvalues · Essential understanding 1

What a matrix is: order, adding and scaling

IB AI AHL 1.14with Prof. Kitso

💻 Using the TI-Nspire CX Student Software on a laptop

The screens in these notes are drawn exactly as the Student Software shows them. The keypad on the left of the software works like the handheld — click its keys with the mouse.
menu: click menu on the on-screen keypad (or Documents Toolbox ▸ 🔧). The menu numbers and letters are the same as in these notes.
enter = Enter · tab = Tab (next box) · esc = Esc · del = Backspace.
Powers: type ^ (Shift+6), the power, then press → to come back down before typing more. x² = ^2 then →.
Fraction: type / · √: type sqrt( · π: type pi · eˣ: type exp( · store: type :=. For EE (×10ⁿ) and i, click them on the keypad.
New page: Ctrl+I (or the Insert menu) · change page: Ctrl+← / Ctrl+→ · graph entry line: Ctrl+G · var: Ctrl+L.
Radian / Degree: click Settings in the status bar at the bottom of the window. The mode (RAD or DEG) shows in the top bar.
IB exam mode (Press-to-Test): everything in these notes works on the non-CAS TI-Nspire with the IB exam settings. In exam mode there are no dot-product, cross-product, unit-vector or eigenvalue keys and no Diff Eq graphs — the notes show the allowed way (type the components, use polyRoots on the characteristic equation, do Euler steps on a Calculator page).
If you type nDeriv(, the software shows it as centralDiff( — it is the same command.

The big idea: A matrix is a neat rectangle of numbers. Its order is rows × columns. You can add or subtract two matrices only if they have the same order, and you can multiply by a number by scaling every entry.

By the end of this lesson you can:

write down the order of a matrix and pick out an entry such as $a_{23}$
recognise square, zero and identity matrices
add, subtract and scale matrices, and say when it is impossible
use equal matrices to find unknown entries
solve simple matrix equations such as $3X - A = B$
enter matrices on your TI-Nspire CX II

🧑🏾 Imagine this…

A shop in Francistown keeps a stock table: three rows (maize meal, rice, sugar) and two columns (Monday, Tuesday). You already read tables like this every day. A matrix is just that table with the labels removed and big round brackets put around the numbers. Once the numbers are in a matrix, you can do arithmetic on the whole table at once.

Matrix

a rectangular array of numbers (called entries or elements) written inside brackets.

Order

the size of a matrix: rows × columns. A matrix with 3 rows and 4 columns has order

3 \times 4

. Always say rows first.

Entry

a_{ij}

the number in row i, column j. So

a_{23}

is in row 2, column 3.

Square matrix

a matrix with the same number of rows and columns, e.g.

2 \times 2

3 \times 3

Zero matrix

every entry is 0. It is written

\mathbf{0}

(or

O

). Adding it changes nothing.

Identity matrix

a square matrix with 1s down the main diagonal (top-left to bottom-right) and 0s everywhere else, e.g.

\begin{pmatrix}1&0\\0&1\end{pmatrix}

. It is written

I

. You will see why it matters in Lesson 2.

★ The three rules

Add or subtract: the matrices must have the same order. Add or subtract entries in the same position.
Scalar multiple: $kA$ means multiply every entry of $A$ by the number $k$ . Any order is allowed.
Equal matrices: $A = B$ only if they have the same order and every pair of matching entries is equal. This gives one equation for each entry.
Rules that behave like ordinary numbers: $A + B = B + A$ , $(A + B) + C = A + (B + C)$ , $k(A + B) = kA + kB$ , $A + \mathbf{0} = A$ , and $A - A = \mathbf{0}$ .

⚠️ Mistakes that lose marks

Order is rows × columns, never columns × rows. A $2 \times 3$ matrix has 2 rows.
Matrices of different orders cannot be added. Write “not possible, the orders are different” and give both orders.
When you subtract, watch the signs: $5 - (-3) = 8$ .
Do not write a matrix with straight lines |...|. Those mean the determinant (Lesson 3), which is a single number.

💡 Entering a matrix on your TI-Nspire CX II

On a Calculator page use the Templates key and choose the matrix template. Pick the number of rows and columns, type each entry and press tab💻 Tab to move to the next box. Store it with ctrl💻 Ctrl var💻 Ctrl+L and a letter, e.g. → a. You can then type a+b, 2*a or a-b.

🧪 Try it yourselfCheck your understanding

Four quick questions. Think about the order first, every time.

Worked examples

Example 1.1Order and entries

Let

P =

\begin{pmatrix}4&-1&0&7\\2&5&3&-2\\6&1&-8&9\end{pmatrix}

(a) Write down the order of $P$ .
(b) Write down the entries $p_{23}$ and $p_{34}$ .
(c) In which row and column is the entry $-8$ ?
(d) Is $P$ a square matrix?

🧠 Think it through

Count the rows (going down), then the columns (going across).
$p_{23}$ : go to row 2, then column 3.
Find −8 by looking along each row.
Square means the number of rows equals the number of columns.

✅ Answer (a)

3 \times 4

. (b)

p_{23} = 3

and

p_{34} = 9

. (c) Row 3, column 3, so it is

p_{33}

. (d) No: 3 rows but 4 columns.

In the exam, “write down” means no working is needed. Just count carefully.

Example 1.2Adding and subtracting stock tables

A wholesaler delivers to three shops in Gaborone, Francistown and Maun. The matrices show the bags of maize meal, rice and sugar delivered in Week 1 (

S

) and Week 2 (

T

). Rows are the shops in that order.

S =

\begin{pmatrix}40&25&30\\22&18&15\\35&20&25\end{pmatrix}

and

T =

\begin{pmatrix}45&20&28\\30&15&18\\32&26&22\end{pmatrix}

(a) Find $S + T$ and explain what it shows.
(b) Find $T - S$ and explain what the negative entries mean.
(c) A new matrix $U$ has order $2 \times 3$ . Explain why $S + U$ cannot be found.

🧠 Think it through

Both $S$ and $T$ are $3 \times 3$ , so adding is allowed.
Add entries in the same position: first row, first column with first row, first column, and so on.
A negative entry in $T - S$ means Week 2 was smaller than Week 1.
Compare orders: $3 \times 3$ and $2 \times 3$ .

✍️ By hand

S =

\begin{pmatrix}40&25&30\\22&18&15\\35&20&25\end{pmatrix}

T =

\begin{pmatrix}45&20&28\\30&15&18\\32&26&22\end{pmatrix}

S + T =

\begin{pmatrix}40+45&25+20&30+28\\22+30&18+15&15+18\\35+32&20+26&25+22\end{pmatrix}

=

\begin{pmatrix}85&45&58\\52&33&33\\67&46&47\end{pmatrix}

T - S =

\begin{pmatrix}5&-5&-2\\8&-3&3\\-3&6&-3\end{pmatrix}

🔢 On your TI-Nspire CX II

On a Calculator page store both matrices: type each one with the matrix template, then ctrl💻 Ctrl var💻 Ctrl+L and the names s1 and t1.
Type s1+t1 enter ↵💻 Enter, then t1-s1 enter ↵💻 Enter.

✅ Answer (a)

S + T =

\begin{pmatrix}85&45&58\\52&33&33\\67&46&47\end{pmatrix}

: the total bags delivered to each shop over the two weeks. (b)

T - S =

\begin{pmatrix}5&-5&-2\\8&-3&3\\-3&6&-3\end{pmatrix}

: for example Gaborone received 5 more bags of maize meal but 5 fewer bags of rice in Week 2. (c)

S

3 \times 3

and

U

2 \times 3

. The orders are different, so

S + U

does not exist.

Give each stored matrix its own short name. Names such as s1 and t1 avoid clashes with the letters the calculator already uses.

Example 1.3Scalar multiples

(a) The monthly prices (in pula) of two services at two lodges are in

P =

\begin{pmatrix}12.5&8\\30&15.2\end{pmatrix}

. All prices rise by 10%. Write the new price matrix.
(b) Given

A =

\begin{pmatrix}2&-1\\0&4\end{pmatrix}

and

B =

\begin{pmatrix}1&3\\-2&5\end{pmatrix}

, find

3A - 2B

🧠 Think it through

A 10% increase means multiply every price by 1.1. That is a scalar multiple, $1.1P$ .
For $3A - 2B$ , scale each matrix first, then subtract entry by entry.

✍️ By hand

1.1P =

\begin{pmatrix}1.1\times12.5&1.1\times8\\1.1\times30&1.1\times15.2\end{pmatrix}

=

\begin{pmatrix}13.75&8.8\\33&16.72\end{pmatrix}

3A =

\begin{pmatrix}6&-3\\0&12\end{pmatrix}

2B =

\begin{pmatrix}2&6\\-4&10\end{pmatrix}

3A - 2B =

\begin{pmatrix}6-2&-3-6\\0+4&12-10\end{pmatrix}

=

\begin{pmatrix}4&-9\\4&2\end{pmatrix}

🔢 On your TI-Nspire CX II

Store $A$ and $B$ , then type 3*a-2*b enter ↵💻 Enter.

✅ Answer (a)

\begin{pmatrix}13.75&8.8\\33&16.72\end{pmatrix}

(b)

\begin{pmatrix}4&-9\\4&2\end{pmatrix}

Money answers are exact to 2 decimal places, so no rounding is needed in (a).

Example 1.4Equal matrices: finding unknowns

Find the values of

x

and

y

such that

2

\begin{pmatrix}x&3\\1&y\end{pmatrix}

+

\begin{pmatrix}1&-2\\0&4\end{pmatrix}

=

\begin{pmatrix}7&4\\2&10\end{pmatrix}

🧠 Think it through

Do the left side first: scale, then add. You get a single matrix with $x$ and $y$ inside.
Two matrices are equal when matching entries are equal. Write one equation per position.
Solve for $x$ and $y$ , then use the other two positions to check.

✍️ By hand

Left side

=

\begin{pmatrix}2x+1&6-2\\2&2y+4\end{pmatrix}

=

\begin{pmatrix}2x+1&4\\2&2y+4\end{pmatrix}

Top left:

2x + 1 = 7 \Rightarrow x = 3

. Bottom right:

2y + 4 = 10 \Rightarrow y = 3

Check: top right

4 = 4

✓ and bottom left

2 = 2

✓.

✅ Answer

x = 3

and

y = 3

The other two positions did not contain unknowns. They are your free check that you have the matrices set up correctly.

Example 1.5Solving a matrix equation

Given

A =

\begin{pmatrix}5&-2\\1&0\end{pmatrix}

and

B =

\begin{pmatrix}4&8\\-7&6\end{pmatrix}

, find the matrix

X

such that

3X - A = B

🧠 Think it through

Treat it like $3x - 5 = 10$ . Add $A$ to both sides first.
Then divide by 3, which means multiply every entry by $\dfrac{1}{3}$ .
Check by putting your $X$ back into $3X - A$ .

✍️ By hand

3X = B + A =

\begin{pmatrix}4+5&8-2\\-7+1&6+0\end{pmatrix}

=

\begin{pmatrix}9&6\\-6&6\end{pmatrix}

X = \dfrac{1}{3}

\begin{pmatrix}9&6\\-6&6\end{pmatrix}

=

\begin{pmatrix}3&2\\-2&2\end{pmatrix}

Check:

3X - A =

\begin{pmatrix}9&6\\-6&6\end{pmatrix}

-

\begin{pmatrix}5&-2\\1&0\end{pmatrix}

=

\begin{pmatrix}4&8\\-7&6\end{pmatrix}

= B

✓

🔢 On your TI-Nspire CX II

Store $A$ and $B$ . Type (b+a)/3 enter ↵💻 Enter.

✅ Answer

X =

\begin{pmatrix}3&2\\-2&2\end{pmatrix}

You can add, subtract and scale matrices on both sides of an equation. What you cannot do (yet) is divide by a matrix. That needs the inverse in Lesson 3.

Challenge questions

⭐ Challenge 1.6Finding kPaper 1 style

Let

P =

\begin{pmatrix}3&k\\2k&1\end{pmatrix}

and

Q =

\begin{pmatrix}1&4\\0&-2\end{pmatrix}

. Given that

P + 2Q =

\begin{pmatrix}5&11\\6&-3\end{pmatrix}

, find the value of

k

✏️ Try it on paper and on your calculator first. Then check.

Show the full solution

🧠 Think it through

Work out $P + 2Q$ in terms of $k$ .
Equate the top right entries to get an equation for $k$ .
Check with the bottom left entry. Both must give the same $k$ .

✍️ By hand

P + 2Q =

\begin{pmatrix}3+2&k+8\\2k+0&1-4\end{pmatrix}

=

\begin{pmatrix}5&k+8\\2k&-3\end{pmatrix}

Top right:

k + 8 = 11 \Rightarrow k = 3

. Bottom left:

2k = 6 \Rightarrow k = 3

✓

✅ Answer

k = 3

⭐ Challenge 1.7Phone shop stockPaper 2 style

A phone shop sells three models in two branches (Gaborone and Kasane). Rows are the branches, columns are Models 1, 2 and 3.
Opening stock

A =

\begin{pmatrix}12&8&5\\10&6&9\end{pmatrix}

, delivery

B =

\begin{pmatrix}6&4&3\\5&5&2\end{pmatrix}

, sold

C =

\begin{pmatrix}9&7&4\\8&3&6\end{pmatrix}

(a) Write down the order of $A$ .
(b) Find the closing stock matrix $A + B - C$ .
(c) How many units of Model 2 does the Kasane branch have at the close?
(d) Next month the shop wants twice the closing stock. Find this target matrix.
(e) Why would $A + B$ not be possible if the Kasane branch had left out Model 3 from its table?

✏️ Try it on paper and on your calculator first. Then check.

Show the full solution

🧠 Think it through

Rows × columns.
Add $A$ and $B$ , then subtract $C$ . All three have the same order.
Kasane is row 2, Model 2 is column 2.
Scalar multiple by 2.
Leaving out a column changes the order, so the orders no longer match.

🔢 On your TI-Nspire CX II

Store $A$ , $B$ and $C$ . Type a+b-c enter ↵💻 Enter and 2*(a+b-c) enter ↵💻 Enter.

✅ Answer (a)

2 \times 3

. (b)

\begin{pmatrix}9&5&4\\7&8&5\end{pmatrix}

. (c) 8 units (row 2, column 2). (d)

\begin{pmatrix}18&10&8\\14&16&10\end{pmatrix}

. (e) With a column missing, that matrix would be

2 \times 2

while

B

2 \times 3

. Matrices can only be added when they have the same order.

Next: Lesson 2 · Multiplying matrices

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL HL · Matrices 1 · What a matrix is: order, adding and scaling

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL

What a matrix is: order, adding and scaling

Worked examples

Challenge questions