# What is matrix rank and how do i calculate it?

The rank of a matrix ${\displaystyle A}$ is the number of independent columns of ${\displaystyle A}$. A square matrix is full rank if all of its columns are independent. That is, a square full rank matrix has no column vector ${\displaystyle v_{i}}$ of ${\displaystyle A}$ that can be expressed as a linear combination of the other column vectors. That is, ${\displaystyle v_{j}\neq \Sigma _{i=1,i\neq j}^{n}a_{i}v_{i}}$ for any set of ${\displaystyle a_{i}}$. For example, if one column of ${\displaystyle A}$ is equal to twice another one, then those two columns are linearly dependent (with a scaling factor 2) and thus the matrix would not be full rank.
For Single-input-single-output (SISO) systems, which are the focus of this course, the reachability matrix will always be square; more inputs make it wider (because the width ${\displaystyle B}$ is equal to the number of inputs). In the case of non-square matrices, full rank means that the number of independent vectors is as large as possible.
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