Unit 4 · Topic 4.10

Matrices

A matrix is a rectangle of numbers with dimensions rows by columns.

Practice 1: Procedural and Symbolic Fluency

Beyond the exam. The AP® Precalculus exam does not test this unit. Many classes teach it, and it prepares you for calculus.

Part 1 · Hook

Why this matters

A food truck sells three items on two days. A table of how many it sold, times a list of prices, gives each day's revenue in one step. That rectangular table is a matrix, and the "row times column" rule for multiplying matrices is exactly the arithmetic the food truck owner would do by hand.

Part 2 · Prerequisite check

Quick check before you start

1. What is $\left\langle 2, -1 \right\rangle \cdot \left\langle 4, 1 \right\rangle$ ?

$7$
$9$
$-7$
$3$

Show the answer

Multiply matching components and add: $2(4) + (-1)(1)$ $= 8 - 1$ $= 7$ .

Correct: $7$ :
$9$ :
$-7$ :
$3$ :

2. Solve the system $x + y = 5$ , $x - y = 1$ .

$x = 3$ , $y = 2$
$x = 2$ , $y = 3$
$x = 4$ , $y = 1$
$x = 5$ , $y = 0$

Show the answer

Adding the equations gives $2x = 6$ , so $x = 3$ and $y = 2$ .

Correct: $x = 3$ , $y = 2$ :
$x = 2$ , $y = 3$ :
$x = 4$ , $y = 1$ :
$x = 5$ , $y = 0$ :

Part 3 · See it

See it first

Matrix A with rows (2, -1) and (0, 3) times matrix B with rows (1, 4) and (-2, 1) equals the product with rows (4, 7) and (-6, 3). Row 1 of A and column 2 of B are highlighted, and the entry 7 in row 1, column 2 of the product is boxed: 2 times 4 plus (-1) times 1 = 7. — Each entry of a product is a row of the first matrix times a column of the second: $2(4) + (-1)(1) = 7$ goes in row 1, column 2. LevlPrep original diagram.

Part 4 · Step by step

How it works, step by step

A matrix has $m$ rows and $n$ columnsits dimensions are $m \times n$ , rows first
The number of columns of $A$ equals the number of rows of $B$ $AB$ is defined; an $m \times n$ times an $n \times p$ gives an $m \times p$
Take row $i$ of $A$ and column $j$ of $B$ multiply matching entries and add: that is the entry in row $i$ , column $j$ of $AB$
Swap the order of the factors $BA$ usually differs from $AB$ , or is undefined at all

Part 5 · Key ideas

Key ideas

A matrix is a rectangular array of numbers. An $m \times n$ matrix has $m$ rows and $n$ columns; $a_{ij}$ is the entry in row $i$ , column $j$ .
Add matrices of the same dimensions entry by entry; a number times a matrix multiplies every entry.
Matrix multiplication: entry $(i, j)$ of $AB$ is row $i$ of $A$ times column $j$ of $B$ , multiplied entry by entry and added. It needs columns of $A$ = rows of $B$ .
Order matters: usually $AB \ne BA$ . The identity $I = \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right]$ gives $AI = IA = A$ .

Part 6 · Misconception

A common mistake

The wrong idea: To multiply two matrices, multiply the entries in the same positions.

What actually happens: Each entry of the product combines a whole row with a whole column. Multiplying matching positions is a different operation, and it does not give the revenue, the combined transformation or anything else the product is used for.

Part 7 · Check yourself

Check yourself

Exam-style questions. Anything you miss goes into your review queue.

Table

Three matrices

$A = \left[ \begin{matrix} 3 & -1 \\ 1 & 2 \end{matrix} \right], \quad B = \left[ \begin{matrix} 2 & 0 \\ -1 & 4 \end{matrix} \right], \quad C = \left[ \begin{matrix} 1 & 0 & 2 \\ 3 & -1 & 1 \end{matrix} \right]$

1. What is $AB$ ?

$\left[ \begin{matrix} 7 & -4 \\ 0 & 8 \end{matrix} \right]$
$\left[ \begin{matrix} 6 & -2 \\ 1 & 9 \end{matrix} \right]$
$\left[ \begin{matrix} 6 & 0 \\ -1 & 8 \end{matrix} \right]$
$\left[ \begin{matrix} 7 & -4 \\ 0 & 6 \end{matrix} \right]$

Show the answer

Row 1 of $A$ with each column of $B$ : $3(2) + (-1)(-1) = 7$ and $3(0) + (-1)(4) = -4$ . Row 2: $1(2) + 2(-1) = 0$ and $1(0) + 2(4) = 8$ .

Correct: $\left[ \begin{matrix} 7 & -4 \\ 0 & 8 \end{matrix} \right]$ : Row times column for each of the four entries.
$\left[ \begin{matrix} 6 & -2 \\ 1 & 9 \end{matrix} \right]$ : This is $BA$ ; the order is reversed.
$\left[ \begin{matrix} 6 & 0 \\ -1 & 8 \end{matrix} \right]$ : This multiplies entries in matching positions; use rows times columns.
$\left[ \begin{matrix} 7 & -4 \\ 0 & 6 \end{matrix} \right]$ : Row 2, column 2: $1(0) + 2(4) = 8$ .

2. What is the entry in row 1, column 2 of $BA$ ?

No calculator. Type the exact value (a fraction, π, e or a root is fine).

Show the answer

Row 1 of $B$ is $(2, 0)$ and column 2 of $A$ is $(-1, 2)$ : $2(-1) + 0(2) = -2$ .

Answer: -2

3. Which products are defined? Select all that apply.

$AC$
$BC$
$CA$
$CB$

Show the answer

$A$ and $B$ are $2 \times 2$ ; $C$ is $2 \times 3$ . A product needs the inside numbers to match.

Correct: $AC$ : $(2 \times 2)(2 \times 3)$ : inside $2 = 2$ .
Correct: $BC$ : $(2 \times 2)(2 \times 3)$ : inside $2 = 2$ .
$CA$ : $(2 \times 3)(2 \times 2)$ : inside $3 \ne 2$ .
$CB$ : $(2 \times 3)(2 \times 2)$ : inside $3 \ne 2$ .

4. What are the dimensions of $AC$ ?

$2 \times 3$
$3 \times 2$
$2 \times 2$
$3 \times 3$

Show the answer

$(2 \times 2)(2 \times 3)$ : the outside numbers give $2 \times 3$ .

Correct: $2 \times 3$ : Rows of $A$ by columns of $C$ .
$3 \times 2$ : Rows come first: $A$ has $2$ rows, $C$ has $3$ columns.
$2 \times 2$ : $C$ has $3$ columns, so the product does too.
$3 \times 3$ : $A$ has $2$ rows, so the product does too.

5. What are the dimensions of $\left[ \begin{matrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{matrix} \right]$ ?

$2 \times 3$
$3 \times 2$
$6 \times 1$
$2 \times 2$

Show the answer

Two rows and three columns: $2 \times 3$ , rows first.

Correct: $2 \times 3$ : Rows by columns.
$3 \times 2$ : Rows come first: there are $2$ rows.
$6 \times 1$ : Count rows and columns, not entries.
$2 \times 2$ : There are $3$ columns.

6. What is $\left[ \begin{matrix} 1 & 2 \\ 3 & 4 \end{matrix} \right]\left[ \begin{matrix} 5 \\ -1 \end{matrix} \right]$ ?

$\left[ \begin{matrix} 3 \\ 11 \end{matrix} \right]$
$\left[ \begin{matrix} 5 \\ -4 \end{matrix} \right]$
$\left[ \begin{matrix} 2 \\ 6 \end{matrix} \right]$
$\left[ \begin{matrix} 3 & 11 \end{matrix} \right]$

Show the answer

Row 1: $1(5) + 2(-1) = 3$ . Row 2: $3(5) + 4(-1) = 11$ . A $2 \times 2$ times a $2 \times 1$ is $2 \times 1$ .

Correct: $\left[ \begin{matrix} 3 \\ 11 \end{matrix} \right]$ : Each row times the column.
$\left[ \begin{matrix} 5 \\ -4 \end{matrix} \right]$ : This multiplies matching positions ( $1 \cdot 5$ and $4 \cdot (-1)$ ); use whole rows.
$\left[ \begin{matrix} 2 \\ 6 \end{matrix} \right]$ : This uses columns of the first matrix instead of rows.
$\left[ \begin{matrix} 3 & 11 \end{matrix} \right]$ : The product is $2 \times 1$ , a column, not a row.

Part 8 · Summary

Summary

A matrix is a rectangle of numbers with dimensions rows by columns. Matrices of the same size add entry by entry. A product

AB

is defined when

A

has as many columns as

B

has rows; each entry is a row of

A

times a column of

B

. The order of the factors matters.

Part 9 · Up next

What comes next

Part 10 · Connections

Connections

Next: The Inverse and Determinant of a Matrix (4.11)

Question set for this topic