We want to determine how many elements there are in a $3 \times 3$ matrix. Recall that a matrix is a rectangular arrangement of numbers that can be used to organize data. To find the number of its elements, we must figure out what the notation $3 \times 3$ actually means.

This means that the matrix in question has

3

rows and

3

columns. Let's take a look at the general form of this kind of matrix with

a_{i j}

being the number in row

i

and column

j .

⎣ ⎢ ⎡ a_{11} a_{21} a_{31} a_{12} a_{22} a_{32} a_{13} a_{23} a_{33} ⎦ ⎥ ⎤

There are exactly

9

elements in all

3 \times 3

matrices.

Finding the Number of Elements for Any Matrix

To find a general rule for all matrices, let's look closely at the connection between the numbers in the given dimensions.

Rows 3 \times Columns 3 = Elements 9

We can notice that the number of elements is equal to the number of rows times the number of columns. This fact holds true for not only

3 \times 3

matrices but for all matrices. To visualize this, we will consider counting the number of elements in a matrix as calculating the area of a rectangle with the same dimensions.

We can calculate the number of elements in a matrix simply by multiplying the number of rows by the number of columns. Finally, let's calculate the number of elements for other examples of matrices.

Matrix	Number of Rows	Number of Columns	Number of Elements
$[1 - 3 21]$	$2$	$2$	$2 \times 2 = 4$
$[4153]$	$1$	$3$	$1 \times 3 = 3$
$[0.5 - 3 1 2.7 3 - 0.5]$	$2$	$3$	$2 \times 3 = 6$
$⎣ ⎢ ⎡ 0.5 3 2.7 1 - 3 - 0.5 ⎦ ⎥ ⎤$	$3$	$2$	$3 \times 2 = 6$

Exercise

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