Matrix Dimensions
Matrix dimensions describe the size of a matrix as the number of rows by the number of columns, written as m x n.
Formula
m \times n \ (\text{rows} \times \text{columns})
Definition
The dimensions of a matrix tell you how many rows and columns it has, written "rows $\times$ columns": a matrix with $m$ rows and $n$ columns is an $m \times n$ matrix, with $m \times n$ total entries. Matrix addition requires equal dimensions, and for matrix multiplication $A \times B$, $A$ must be $m \times n$ and $B$ must be $n \times p$, giving a result of size $m \times p$, the inner dimensions must match while the outer dimensions give the result size. The rank of a matrix (dimension of its column space, equivalently row space) is at most $\min(m,n)$, and the rank-nullity theorem states that for $T: F^n \to F^m$ represented by an $m \times n$ matrix, $\text{rank}(T) + \text{nullity}(T) = n$, relating the dimensions of the image and kernel.
Example
A matrix with $3$ rows and $2$ columns, $\begin{bmatrix}1 & 2\\3 & 4\\5 & 6\end{bmatrix}$, is $3\times 2$, while $\begin{bmatrix}7 & 8 & 9 & 10\end{bmatrix}$ is $1\times 4$. If $A$ is $3\times 4$ and $B$ is $4\times 2$, the product $AB$ is defined (inner dimensions $4=4$ match) and produces a $3\times 2$ matrix, but $BA$ is not defined since the inner dimensions ($2$ and $3$) don't match. A $4\times 3$ matrix with rank $2$ has nullity $3 - 2 = 1$: there is a $1$-dimensional solution space to $Ax = 0$.
Key Insight
A memory trick: "rows come before columns" alphabetically, and this is the correct order for stating dimensions; the multiplication rule ($m \times n$ times $n \times p = m \times p$) is the key constraint in matrix algebra. The rank-nullity theorem is analogous to the first isomorphism theorem in group theory, $\dim(\text{domain}) = \dim(\text{image}) + \dim(\text{kernel})$, quantifying how much information is lost by a linear map.