Home page for accesible maths 2 Row operations on matrices 2.2 Row reduction of matrices to echelon form 2.4 Exercises

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2.3. Rank of a matrix

The echelon form of a matrix is not unique, and so at first you might think that the number of non-zero rows depends on which echelon form you choose. But the number of non-zero rows of an echelon form does not change when putting it into reduced echelon form (which is unique); therefore that number does not depend on the echelon form, and it is called the rank.

Definition 2.3.1.

The rank of a matrix is the number of non-zero rows of its echelon form.

Example 2.3.2.

1. (a)
  
  The following matrices are rank 2:
  
  $\begin{pmatrix}2&0\\ 0&3\end{pmatrix},\quad\begin{pmatrix}1&1&1\\ 0&1&0\\ 0&3&0\end{pmatrix},\quad\begin{pmatrix}1&-1&0\\ 0&2&-2\\ 1&0&-1\end{pmatrix}.$
2. (b)
  
  The following matrices are rank 1:
  
  $\begin{pmatrix}0&0\\ 0&3\end{pmatrix},\quad\begin{pmatrix}0&0&0\\ 0&1&0\\ 0&3&0\end{pmatrix},\quad\begin{pmatrix}1&-1&0\\ 0&0&0\\ 1&-1&0\end{pmatrix}.$

Remark 2.3.3.

It is a non-obvious result in linear algebra that the rank of any matrix equals the rank of its transpose; in other words $\operatorname{rank}A=\operatorname{rank}A^{t}$ (we omit the proof). For example, in 2.2, the rank of $A$ is 2. To compute $\operatorname{rank}A^{t}$ , perform row operations:

\begin{pmatrix}1&1&2\\ 2&3&4\\ 3&5&6\end{pmatrix}\xrightarrow{\begin{array}[]{l}R_{2}=r_{2}-2r_{1}\\ R_{3}=r_{3}-3r_{1}\\ R_{3}=r_{3}-2r_{2}\end{array}}\begin{pmatrix}1&1&2\\ 0&1&0\\ 0&0&0\end{pmatrix}.

So the rank of $A^{t}$ is still 2, as expected.