Understanding Matrices

From a Score Table to a Matrix

Tables often organize information into rows and columns. A class score sheet, for example, can place one student in each row and one subject in each column. A matrix uses the same rectangular arrangement to organize numbers for mathematical work.

The numbers inside a matrix are its elements or entries. We usually enclose them in parentheses $()$ or square brackets $[]$. An entry's row and column determine which quantity it represents, so moving it changes its meaning.

Consider the Mathematics and English scores of three students:

Keeping only the numerical entries gives the matrix:

The same matrix can also be written with parentheses:

$80, 75, 70, 95, 95, 75$ are the elements of matrix $A$.

Rows Columns and Elements

Three terms describe a matrix's structure:

1. A row is a horizontal sequence of elements.
2. A column is a vertical sequence of elements.
3. An element is one entry at a particular row and column.

For matrix $A$:

• Row $1$ is $[80 \ 75]$

• Row $2$ is $[70 \ 95]$

• Row $3$ is $[95 \ 75]$

• Column $1$:

  $$\begin{bmatrix} 80 \\ 70 \\ 95 \end{bmatrix}$$

• Column $2$:

  $$\begin{bmatrix} 75 \\ 95 \\ 75 \end{bmatrix}$$

An element of matrix $A$ is written $a_{ij}$. The first index $i$ gives the row, and the second index $j$ gives the column.

Here $m$ is the number of rows and $n$ is the number of columns. The row index $i$ is an integer satisfying $1\le i\le m$. The column index $j$ is an integer satisfying $1\le j\le n$. These bounds keep each index within the matrix.

For the matrix above:

• $a_{11}$ is in row $1$, column $1$, so $a_{11}=80$.
• $a_{12}$ is in row $1$, column $2$, so $a_{12}=75$.
• $a_{21}$ is in row $2$, column $1$, so $a_{21}=70$.
• $a_{22}$ is in row $2$, column $2$, so $a_{22}=95$.
• $a_{31}$ is in row $3$, column $1$, so $a_{31}=95$.
• $a_{32}$ is in row $3$, column $2$, so $a_{32}=75$.

Dimensions of a Matrix

The dimensions of a matrix state its number of rows first and columns second. A matrix with $m$ rows and $n$ columns has dimensions $m \times n$, read "$m$ by $n$."

For matrix $A$:

Matrix $A$ has $3$ rows and $2$ columns, so its dimensions are $3 \times 2$. We may write this as $A_{3 \times 2}$.

Another example:

Suppose a semester attendance table records excused, sick, and unexcused absences:

Its numerical entries form matrix $B$:

Matrix $B$ has $3$ rows and $3$ columns. Its dimensions are therefore $3 \times 3$, written $B_{3 \times 3}$.

For example:

• $b_{11} = 2$ is the element in row $1$, column $1$.
• $b_{23} = 1$ is the element in row $2$, column $3$.
• $b_{32} = 2$ is the element in row $3$, column $2$.

General Matrix Notation

A matrix $A$ with $m$ rows and $n$ columns can be written as:

The notation tells us:

• $A_{m \times n}$ is matrix $A$ with dimensions $m \times n$.
• $m$ is the number of rows.
• $n$ is the number of columns.
• $a_{ij}$ is the element in row $i$ and column $j$.

Matrix types and operations use row and column positions to identify which entries are compared, combined, or transformed.