Chapter 10, Problem 5RCC

To determine

To explain: about the statement A is a matrix with dimension $m\times n$ .

Answer to Problem 5RCC

It means that matrix A has $m^{th}$ row and the $n^{th}$ column.

Explanation of Solution

A matrix is a rectangular array of numbers. The dimension of a matrix indicates how many rows and how many columns it has. A matrix with dimension $m\times n$ has m rows and n columns.

We can index each value in the matrix with a subscript of two numbers. The first of the two numbers is the row number, and the second is the column number. For example, the top left value in the matrix can be called $a_{11}$ , since it is in the first row and column. The bottom right value in the matrix is $a_{mn}$ , since it is in the $m^{th}$ row and the $n^{th}$ column.

An $m\times n$ matrix would look like this.

  $\left[\begin{array}{ccccc}{a}_{11}& a_{12}& a_{13}& \cdots & a_{1n}\\ a_{21}& a_{22}& a_{23}& \cdots & a_{2n}\\ a_{31}& a_{32}& a_{33}& \cdots & a_{3n}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ a_{m1}& a_{m2}& a_{m3}& \cdots & a_{mn}\end{array}\right]$

As a more specific example, say we had a $2\times 3$ matrix filled with integers. It might look like this:

  $\left[\begin{array}{ccc}0& 1& 2\\ 3& 4& 5\end{array}\right]$

Notice that it has $m=2$ rows and $n=3$ columns.