Matrices 

$\left[\begin{matrix}a_1&a_2&a_3\\b_1&b_2&b_3\end{matrix}\right]$
This is a matrix with two rows and three columns  it is a 2x3 matrix.
Adding matrices is easy  it only works if they're the same size, and you do it entry by entry.
Multiplying is much less obvious, but arises naturally by thinking of a matrix as a linear transformation from $R^n$ to $R^m.$ Thinking of matrix multiplication in that way makes it clear why division of matrices in not generally defined, but the inverse of a matrix will sometimes (but not always) exist.
Specifically, think of a matrix as a mapping from $R^n$ to $R^m$ and consider the space in $R^m$ of all points that can be hit. If the dimension of that space is n, then the mapping can be undone. That means the mapping has an inverse, and so the matrix has an inverse.
More later ...