# 2.7 Rank of a Matrix

The rank of an n × n matrix A is the maximum number of linearly independent columns in A. Column vectors of of A, denoted ai, are linearly independent if the only set of scalars $\alpha_{i}$ such that

 ${\alpha}_{1}\mathbf{a^{1}}+{\alpha}_{2}\mathbf{a^{2}}+...+{\alpha}_{n}\mathbf{% a^{n}}=\mathbf{0}$ (7)

is the set

 ${\alpha}_{1}={\alpha}_{2}=...={\alpha}_{n}=0$

For a more concrete example, consider the following matrix.

 $\mathbf{A}=\left(\begin{array}[]{cccc}0&1&1&2\\ 1&2&3&4\\ 2&0&2&0\end{array}\right)$

The rank of A is two, since its third and fourth columns are linear combinations of its first two columns, i.e.

 $\displaystyle\mathbf{{a}^{3}}$ $\displaystyle=\mathbf{{a}^{1}+{a}^{2}}$ $\displaystyle\mathbf{{a}^{4}}$ $\displaystyle=2\mathbf{{a}^{2}}$

If A is an n × n matrix, it can be shown

 $\text{rank}\left(\mathbf{A}\right)\leq\text{min}(m,n)$ (8)

Furthermore,

 $\text{rank}\left(\mathbf{AB}\right)\leq\text{ min}(\text{rank}\left(\mathbf{A}% \right),\text{rank}\left(\mathbf{B}\right))$ (9)

and

 $\text{rank}\left(\mathbf{A{A}^{T}}\right)=\text{rank}\left(\mathbf{{A}^{T}A}% \right)=\text{rank}\left(\mathbf{A}\right)$ (10)