Matrix Dimensions
Functions & Advanced AlgebraMatrix 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. We write it as "rows $\times$ columns." A $3\times 4$ matrix has $3$ rows and $4$ columns.
Example
$\begin{bmatrix}1 & 2\\3 & 4\\5 & 6\end{bmatrix}$ has $3$ rows and $2$ columns: it is a $3\times 2$ matrix. A $1\times 4$ matrix: $\begin{bmatrix}7 & 8 & 9 & 10\end{bmatrix}$ has $1$ row and $4$ columns.
Key Insight
Always say rows first, then columns. A memory trick: "rows come before columns" alphabetically in "rc," and "rows-columns" is the correct order for dimensions.
Definition
A matrix with $m$ rows and $n$ columns is called an $m \times n$ matrix. The total number of entries is $m \times n$. Matrix addition requires equal dimensions. For matrix multiplication $A \times B$, $A$ must be $m \times n$ and $B$ must be $n \times p$; the result is $m \times p$.
Example
$A$ is $3\times 4$, $B$ is $4\times 2$. The product $AB$ is defined (inner dimensions match: $4=4$) and produces a $3\times 2$ matrix. The product $BA$ is not defined ($4\times 2$ times $3\times 4$: inner $2 \neq 3$).
Key Insight
The dimension rule for multiplication ($m \times n$ times $n \times p = m \times p$) is the key constraint in matrix algebra. The inner dimensions must match; the outer dimensions give the result size.
Definition
The dimension of $M_{m,n}(F)$ as a vector space is $mn$. The rank of a matrix (dimension of its column space, equivalently row space) is at most $\min(m,n)$. The rank-nullity theorem: 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 $4\times 3$ matrix $A$ has rank at most $3$. If $\text{rank}(A) = 2$, then $\text{nullity}(A) = 3 - 2 = 1$: there is a $1$-dimensional solution space to $Ax = 0$.
Key Insight
The rank-nullity theorem is a fundamental result in linear algebra, analogous to the first isomorphism theorem in group theory: $\dim(\text{domain}) = \dim(\text{image}) + \dim(\text{kernel})$. It quantifies how much information is lost by a linear map.