The rank of a matrix is the number of [[Linearly Independent Vectors|linearly independent]] rows or columns in the matrix. It provides important information about the dimensionality of the vector space spanned by the rows or columns. - The rank is always non-negative. - A matrix with all zero entries ("zero matrix") has rank $0$. - For any matrix $A$, the rank of its rows is equal to the rank of its columns. **Full Rank:** - A matrix has full rank if its rank is equal to the smaller of the number of rows or columns. - If $A$ is an $n \times d$ matrix, then: $\text{rank}(A) = \min(n,d) $ **Example:** In the following matrix the first row is $-2$ times the second row $(R_2=-2R_1)$. Therefore only 1 linearly independent row remains. $ A = \begin{bmatrix} \phantom{-}3 & \phantom{-}2 & \phantom{-}1 \\ -6 & -4 & -2 \end{bmatrix} \implies \text{rank}(A)=1 $ The computation can also be done row-wise, as both approaches always lead to the same result.