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

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The rank of a matrix $A$ is the number of independent columns of $A$. A square matrix is full rank if all of its columns are independent. That is, a square full rank matrix has no column vector $v_{i}$ of $A$ that can be expressed as a linear combination of the other column vectors. That is, $v_{j}\neq \Sigma _{{i=1,i\neq j}}^{{n}}a_{i}v_{i}$ for any set of $a_{i}$. For example, if one column of $A$ 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 $B$ 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.