Matrices — AP® Precalculus Notes

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

A matrix (plural matrices) is a rectangle of numbers arranged in rows and columns. Matrices organize data, like sales by day and item, and later in this unit they turn into functions that move, rotate and stretch the plane. This page covers the language and the arithmetic, above all how to multiply.

Dimensions and entries

A matrix with $m$ rows and $n$ columns has dimensions $m \times n$ (say "m by n"), rows first. The entry in row $i$ and column $j$ is written $a_{ij}$ .

A = \left[ \begin{matrix} 3 & -1 & 4 \\ 0 & 2 & 5 \end{matrix} \right]

is $2 \times 3$ , and $a_{13} = 4$ (row 1, column 3). A matrix with one column, such as $\left[ \begin{matrix} 5 \\ -1 \end{matrix} \right]$ , is a column vector.

Adding and scaling

Matrices with the same dimensions add entry by entry, and a number times a matrix multiplies every entry:

\left[ \begin{matrix} 1 & -2 \\ 0 & 4 \end{matrix} \right] + \left[ \begin{matrix} 3 & 1 \\ 2 & -1 \end{matrix} \right] = \left[ \begin{matrix} 4 & -1 \\ 2 & 3 \end{matrix} \right], \qquad 3\left[ \begin{matrix} 1 & -2 \\ 0 & 4 \end{matrix} \right] = \left[ \begin{matrix} 3 & -6 \\ 0 & 12 \end{matrix} \right].

Multiplying: row times column

The product $AB$ is defined when the number of columns of $A$ equals the number of rows of $B$ . If $A$ is $m \times n$ and $B$ is $n \times p$ , then $AB$ is $m \times p$ : the outside numbers give the size, the inside numbers must match.

To find the entry in row $i$ , column $j$ of $AB$ , take row $i$ of $A$ and column $j$ of $B$ , multiply matching entries and add, just like a dot product.

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. — **Figure 1.** Row 1 of $A$ times column 2 of $B$ gives the entry in row 1, column 2 of $AB$ . LevlPrep original diagram.

Worked example 1. Find $AB$ for $A = \left[ \begin{matrix} 2 & -1 \\ 0 & 3 \end{matrix} \right]$ and $B = \left[ \begin{matrix} 1 & 4 \\ -2 & 1 \end{matrix} \right]$ .

Row 1, column 1: $2(1) + (-1)(-2)$ $= 2 + 2$ $= 4$ . Row 1, column 2: $2(4) + (-1)(1) = 7$ .

Row 2, column 1: $0(1) + 3(-2) = -6$ . Row 2, column 2: $0(4) + 3(1) = 3$ .

So $AB = \left[ \begin{matrix} 4 & 7 \\ -6 & 3 \end{matrix} \right]$ . In the other order, $BA = \left[ \begin{matrix} 2 & 11 \\ -4 & 5 \end{matrix} \right]$ : a different matrix. Order matters.

Matrices in context

Worked example 2. A food truck sells burgers, fries and drinks at $6, $3 and $2. On Saturday it sells $40$ , $65$ and $90$ ; on Sunday $55$ , $50$ and $70$ . Find each day's revenue.

Put the sales in a $2 \times 3$ matrix (one row per day) and the prices in a $3 \times 1$ column:

\left[ \begin{matrix} 40 & 65 & 90 \\ 55 & 50 & 70 \end{matrix} \right]\left[ \begin{matrix} 6 \\ 3 \\ 2 \end{matrix} \right] = \left[ \begin{matrix} 40(6) + 65(3) + 90(2) \\ 55(6) + 50(3) + 70(2) \end{matrix} \right] = \left[ \begin{matrix} 615 \\ 620 \end{matrix} \right].

Saturday brought in $615 and Sunday $620. The inside dimensions match ( $3$ and $3$ ) and the result is $2 \times 1$ , one revenue per day.

Rules to keep straight

Operation	When it is defined	Result
$A + B$	same dimensions	same dimensions, entry by entry
$kA$	any matrix	every entry times $k$
$AB$	columns of $A$ = rows of $B$	rows of $A$ by columns of $B$

The identity matrix $I = \left[ \begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix} \right]$ plays the role of the number $1$ : $AI = IA = A$ for every $2 \times 2$ matrix $A$ . Unlike numbers, though, $AB$ and $BA$ are usually different, so always multiply in the order the problem gives.