The i's and j's in a Matrix

Question

I know that i means row and j means column, what i don't understand is what are they meaning when they say that the row is greater than or = to 1? And the column is less than or equal to 3? I don't understand what is being said.

Please you maths boffins out there! Help!

Thank you in advance!

Answer 1

I'll focus on why what $i$ and $j$ can be is restricted actually makes sense in the context of matrices, rather than just the specific statement made in your question.

The rows and columns are indexed by $i$ and $j$, respectively. But there are only a finite number of rows and columns, so it can't be that every pair of natural numbers $i,j$ is meant. Thus, they need to be specified, and this specification is precisely $1\leq i,j\leq 3$.

Specifically, an $n\times m$ matrix with entries in $\mathbb{R}$ is a map $\mathbf{A}:\{1,\ldots, n\}\times \{1,\ldots, m\}\rightarrow \mathbb{R}$, so specifying what $i$ and $j$ can range over gives the domain of this function.

Answer 2

When you say $a_{ij}$, you mean the element in the $i^{th}$ row and $j^{th}$ column. So i means row index while j means column index. When said that $1<=i,j<=3$, it only means that i or j can atmost vary between $1$ and $3$.

Answer 3

instead of saying

$$a_{11},a_{12},a_{13},a_{21},a_{22},a_{23},a_{31},a_{32},a_{33}\in \mathbb{R}$$

it said

$a_{ij}\in \mathbb{R}$ for each $1\leq i,j\leq 3$