Real or Complex Matrix Entries and Dimension

A matrix is a rectangular (or square) box of real (or complex) numbers.  

For example:  

$$A = \begin{bmatrix} 1 & 2 \\ 7 & -i \end{bmatrix}, \quad B = \begin{bmatrix} a & b & c \\ d & e & f \end{bmatrix}, \quad C = \begin{bmatrix} 1 & 4 \\ -2 & 3 \\ \pi & e^\frac{-i \pi}{4} \\ 7 & 5 \end{bmatrix}$$

We say the first matrix $A$ has dimension $2 \times 2$, i.e. 2 rows and 2 columns. We also say $A$ is a square matrix.

Thus, $B$ has dimension $2 \times 3$ because it has 2 rows and 3 columns.

And $C$ has dimension $4 \times 2$ because it has 4 rows and 2 columns.

In general, if a matrix has $m$ rows and $n$ columns, we say it is an $m \times n$ matrix, or that it has dimension $m \times n$.

It is a natural question to ask: what is the point of these boxes of numbers?

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