Matrices and their order

A matrix is a rec­tan­gu­lar arrange­ment of num­bers in rows and columns. It is a com­pact way to store infor­ma­tion that comes in a grid, such as the prices of sev­eral items in sev­eral shops, and it is also a pow­er­ful tool for solv­ing sys­tems of lin­ear equa­tions, which is where this course ends. This first les­son explains what a matrix is, how to describe its size, how to name its entries, and how to build a matrix from a rule.

What a matrix is#

Sup­pose two shops in Vijayawada sell rice, dal and oil. The price per kg (in rupees) can be set out in a grid:

If every­one agrees that the rows are shops and the columns are items, the labels can be dropped, leav­ing just the num­bers in brack­ets:

$$\begin{bmatrix} 52 & 110 & 180 \\ 60 & 104 & 170 \end{bmatrix}$$

Def­i­n­i­tion. A matrix is a rec­tan­gu­lar array of num­bers arranged in rows and columns and enclosed in brack­ets. The num­bers are called the ele­ments or entries of the matrix.

Matri­ces are usu­ally named with cap­i­tal let­ters: $A$, $B$, $P$, and so on.

• A row is a hor­i­zon­tal line of entries. Rows are counted from the top.
• A col­umn is a ver­ti­cal line of entries. Columns are counted from the left.

The order of a matrix#

A matrix with $m$ rows and $n$ columns is said to have order $m \times n$ (read "$m$ by $n$"). It has $m \times n$ ele­ments in all.

The price matrix above has $2$ rows and $3$ columns, so its order is $2 \times 3$, and it has $6$ ele­ments.

The order always gives rows first, then columns. A $2 \times 3$ matrix and a $3 \times 2$ matrix are dif­fer­ent shapes:

$$\begin{bmatrix} 1 & 4 & 0 \\ -2 & 7 & 5 \end{bmatrix} \text{ is } 2 \times 3, \qquad \begin{bmatrix} 1 & -2 \\ 4 & 7 \\ 0 & 5 \end{bmatrix} \text{ is } 3 \times 2$$

Exam­ple 1#

A matrix has $12$ ele­ments. What are its pos­si­ble orders?

The order $m \times n$ must have $m \times n = 12$, with $m$ and $n$ pos­i­tive inte­gers. The fac­tor pairs of $12$ give six orders:

$$1 \times 12, \quad 2 \times 6, \quad 3 \times 4, \quad 4 \times 3, \quad 6 \times 2, \quad 12 \times 1$$

Nam­ing the ele­ments#

The ele­ment in row $i$ and col­umn $j$ of a matrix $A$ is writ­ten $a_{ij}$. The first sub­script is the row, the sec­ond is the col­umn. So a $2 \times 3$ matrix looks like

$$A = \begin{bmatrix} a_{11} & a_{12} & a_{13} \\ a_{21} & a_{22} & a_{23} \end{bmatrix}$$

In short, we write $A = [a_{ij}]_{m \times n}$.

Exam­ple 2#

For $A = \begin{bmatrix} 2 & -1 & 4 \\ 0 & 5 & 7 \\ 3 & 8 & -6 \end{bmatrix}$, write the order and the ele­ments $a_{12}$, $a_{23}$, $a_{31}$ and $a_{33}$.

$A$ has $3$ rows and $3$ columns, so its order is $3 \times 3$.

• $a_{12}$: row $1$, col­umn $2$, which is $-1$.
• $a_{23}$: row $2$, col­umn $3$, which is $7$.
• $a_{31}$: row $3$, col­umn $1$, which is $3$.
• $a_{33}$: row $3$, col­umn $3$, which is $-6$.

Build­ing a matrix from a rule#

Some­times a matrix is described by a for­mula for $a_{ij}$. To build it, work out each entry from its row num­ber $i$ and col­umn num­ber $j$.

Exam­ple 3#

Con­struct the $2 \times 3$ matrix with $a_{ij} = i + 2j$.

There are $2$ rows ($i = 1, 2$) and $3$ columns ($j = 1, 2, 3$).

Row 1: $a_{11} = 1 + 2 = 3$, $a_{12} = 1 + 4 = 5$, $a_{13} = 1 + 6 = 7$.

Row 2: $a_{21} = 2 + 2 = 4$, $a_{22} = 2 + 4 = 6$, $a_{23} = 2 + 6 = 8$.

$$A = \begin{bmatrix} 3 & 5 & 7 \\ 4 & 6 & 8 \end{bmatrix}$$

Exam­ple 4#

Con­struct the $3 \times 2$ matrix with $a_{ij} = 2i - j$.

Row 1 ($i = 1$): $2 - 1 = 1$ and $2 - 2 = 0$.

Row 2 ($i = 2$): $4 - 1 = 3$ and $4 - 2 = 2$.

Row 3 ($i = 3$): $6 - 1 = 5$ and $6 - 2 = 4$.

$$A = \begin{bmatrix} 1 & 0 \\ 3 & 2 \\ 5 & 4 \end{bmatrix}$$

Read­ing a matrix in con­text#

Exam­ple 5#

Sita runs a sta­tionery stall. Her stock at the start of the week is: $40$ pens, $25$ note­books and $10$ geom­e­try boxes in her first stall; $35$ pens, $30$ note­books and $12$ geom­e­try boxes in her sec­ond stall. Write this as a matrix with stalls as rows, and say what $a_{23}$ means.

$$S = \begin{bmatrix} 40 & 25 & 10 \\ 35 & 30 & 12 \end{bmatrix}$$

The order is $2 \times 3$. The entry $a_{23} = 12$ is in row $2$ (the sec­ond stall) and col­umn $3$ (geom­e­try boxes): the sec­ond stall has $12$ geom­e­try boxes.

Prac­tice#

1. Write the order of $\begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{bmatrix}$ and the num­ber of its ele­ments.
2. For $A = \begin{bmatrix} 2 & -1 & 4 \\ 0 & 5 & 7 \\ 3 & 8 & -6 \end{bmatrix}$, write $a_{21}$, $a_{32}$ and $a_{13}$.
3. Write all the pos­si­ble orders of a matrix with $8$ ele­ments.
4. Write all the pos­si­ble orders of a matrix with $13$ ele­ments.
5. Con­struct the $2 \times 2$ matrix with $a_{ij} = ij$.
6. Con­struct the $2 \times 3$ matrix with $a_{ij} = i - j$.
7. Con­struct the $3 \times 2$ matrix with $\displaystyle a_{ij} = \frac{(i + j)^2}{2}$.
8. Con­struct the $3 \times 3$ matrix with $a_{ij} = 2i + j$.
9. In Exam­ple 5, which entry gives the num­ber of note­books in the first stall, and what is it?

Answers#