Rank of a Matrix

To define rank precisely, recall: a matrix formed by deleting some rows and some columns of $A$ is a sub-matrix of $A$ (a matrix is trivially a sub-matrix of itself, deleting zero rows and zero columns), and the determinant of a square sub-matrix is called a minor of $A$.

Definition 1.6 (rank). The rank of a matrix $A$, written $\rho(A)$, is the order of the largest square sub-matrix of $A$ whose determinant is non-zero -- equivalently, the largest $r$ such that some $r\times r$ minor of $A$ is non-zero while every minor of order $r+1$ (and higher, if any) vanishes. The rank of the zero matrix is defined to be $0$.

Basic facts. (i) If $A$ has at least one non-zero entry, $\rho(A)\ge1$. (ii) $\rho(I_n)=n$. (iii) If $\rho(A)=r$, some order-$r$ minor is non-zero and every order-$(r{+}1)$ minor (if any exist) vanishes. (iv) For an $m\times n$ matrix, $\rho(A)\le\min\{m,n\}$. (v) A square matrix $A$ of order $n$ has an inverse if and only if $\rho(A)=n$.

Worked illustration (minor method). For $A=\begin{pmatrix}1&2&3\\2&4&7\\3&6&10\end{pmatrix}$: the sole $3\times3$ minor is $|A|=1(40-42)-2(20-21)+3(12-12)=-2+2+0=0$, so $\rho(A)<3$. The $2\times2$ minor $\begin{vmatrix}1&3\\2&7\end{vmatrix}=7-6=1\ne0$, so $\rho(A)=2$.

Theorem 1.11. The rank of a matrix in row-echelon form equals its number of non-zero rows.

Theorem 1.12. The rank of any non-zero matrix equals the number of non-zero rows in a row-echelon form of it -- since elementary transformations never change rank (row/column swaps only relabel minors up to sign; scaling multiplies a minor by a non-zero constant, never turning it zero or vice versa; and the add-a-multiple operation is exactly the one used to build the echelon form in the first place). …