Each column corresponds to a size (S, M, L, XL) and each row
corresponds to a product (T-shirts, pants, sweaters).
Size
The entries of a matrix are called elements.
We say that a matrix with $n$ rows and $m$
columns is $n \times m$.
We refer to the element in the $i$th row and
$j$th column of the matrix $A$ by
$a_{\color{var(--emphColor)}{ij}}$.
By convention the row index always comes first!
As an array of elements
A matrix is determined by its elements, so specifying an
$\color{var(--emphColor)}{n} \times \color{var(--emphColor)}{m}$
matrix $A$ is the same thing as specifying each $a_{ij}$, for
$1 \leq i \leq \color{var(--emphColor)}{n}$ and
$1 \leq j \leq \color{var(--emphColor)}{m}$.
Matrix math
Matrices are algebraic objects and as such we can perform
operations with/on them.
(New inventory) Suppose 1 large T-shirt, 3 small pants, 4 large
pants and 7 medium sweaters are shipped to the store. Which matrix defines the new inventory?
Matrix addition
(New inventory) Suppose 1 large T-shirt, 3 small pants, 4 large
pants and 7 medium sweater are shipped to the store. Which matrix defines the new inventory?
Arrange the new inventory into a matrix and add it to the old inventory:
\begin{equation}
\begin{bmatrix}
5 & 3 & 0 & 0\\
0 & 0 & 4 & 0\\
1 & 0 & 10 & 2\\
\end{bmatrix}
+ \begin{bmatrix}
0 & 0 & 1 & 0\\
3 & 0 & 4 & 0\\
0 & 7 & 0 & 2\\
\end{bmatrix}
=
\begin{bmatrix}
5 & 3 & 1 & 0\\
3 & 0 & 8 & 0\\
1 & 7 & 10 & 2\\
\end{bmatrix}.
\end{equation}
Since we've arranged both arrays in the same way, we may simply
add corresponding entries together to obtain the correct result!
Addition requires the arrays to have the same number of rows and columns,
and we compute it component-wise: each entry in the sum array is the sum of
corresponding entries in the summands.
Matrix Addition
Our motivating example suggests the following definition.
If $A$ and $B$ are $\color{var(--emphColor)}{n \times m}$ matrices, then the sum $A + B$ is an
$\color{var(--emphColor)}{n \times m}$ matrix given by
$$(A+B)_{ij} = a_{ij} + b_{ij}.$$