Matrices: representing data, adding and multiplying by a scalar

Discover

A table has 2 rows and 3 columns of numbers. How many numbers does it hold?

6
5

Key idea

A matrix is a rectangular table of numbers.

An

m \times n

matrix has

m

rows and

n

columns; the entry in row

i

, column

j

is

a_{ij}

. Matrices of the same size add and subtract entry by entry, and a scalar

k

multiplies every entry:

3\begin{pmatrix} 1 & 2 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 3 & 6 \\ 0 & -3 \end{pmatrix}

.

Worked example

Add $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $\begin{pmatrix} 5 & 0 \\ -1 & 2 \end{pmatrix}$ .

1 + 5, 2 + 0, 3 + (−1), 4 + 2

Same positions.

\begin{pmatrix} 6 & 2 \\ 2 & 6 \end{pmatrix}

Result.

Your turn, guided

Your turn: the top-left entry of 4 $\begin{pmatrix} 2 & 1 \\ 0 & 3 \end{pmatrix}$ .

8

4 · 2.

Practice ★

Let A = $\begin{pmatrix} 3 & -3 \\ 7 & 5 \end{pmatrix}$ and B = $\begin{pmatrix} -4 & -4 \\ -5 & 5 \end{pmatrix}$ . What is the entry in row 1, column 2 of A + B?
Let A = $\begin{pmatrix} 2 & 7 \\ 6 & 1 \end{pmatrix}$ and B = $\begin{pmatrix} 5 & -5 \\ 5 & 5 \end{pmatrix}$ . What is the entry in row 1, column 2 of A + B?

A shop records sales in a matrix with one row for each of its 4 stores and one column for each of 7 days. What are the dimensions of the matrix?
1. 4 × 7
2. 7 × 4
3. 28 × 1

Watch out!

Practice ★★

Let A = $\begin{pmatrix} 2 & 0 \\ 6 & 2 \end{pmatrix}$ . What is the entry in row 2, column 1 of 3A?
Let A = $\begin{pmatrix} 8 & 2 \\ -1 & -3 \end{pmatrix}$ . What is the entry in row 2, column 1 of 3A?

Let A = $\begin{pmatrix} 3 & -3 \\ 7 & 5 \end{pmatrix}$ and B = $\begin{pmatrix} -4 & -4 \\ -5 & 5 \end{pmatrix}$ . What is the entry in row 1, column 2 of A + B?

Exit ticket

Three questions to check you have got it.

Let A = $\begin{pmatrix} 3 & -3 \\ 7 & 5 \end{pmatrix}$ and B = $\begin{pmatrix} -4 & -4 \\ -5 & 5 \end{pmatrix}$ . What is the entry in row 1, column 2 of A + B?
Let A = $\begin{pmatrix} 2 & 7 \\ 6 & 1 \end{pmatrix}$ and B = $\begin{pmatrix} 5 & -5 \\ 5 & 5 \end{pmatrix}$ . What is the entry in row 1, column 2 of A + B?
Let A = $\begin{pmatrix} 4 & 6 \\ 1 & 8 \end{pmatrix}$ and B = $\begin{pmatrix} 7 & 2 \\ 5 & 9 \end{pmatrix}$ . What is the entry in row 1, column 2 of A + B?

Summary card

Matrices

Size is rows × columns; add entry by entry; a scalar multiplies every entry.

Watch out: rows first.

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