I am a novice in the field of matrices, and I am doing some self-learning. I would like to ask what might be a very basic question, but I would be very appreciative of your help.
Singular matrix is defined as a square matrix with the determinant of zero. The determinant of zero occurs when matrix columns are linearly dependent (i.e. one of the columns can be defined as a linear combination of other columns).
However, some sources also note that the determinant can be zero when there is linear dependency not only among the columns but also among the rows of a square matrix. Therefore, I wanted to ask the following in the context of the singular matrices:
Does linear dependency among columns imply that there is automatically linear dependency among rows of data?
Or, does linear dependency result from only columns, or only rows, or both?
Note: I originally posted this question on Cross-validated, but I did not get an answer. So I thought that maybe this network is more appropriate for this type of question.
Best Answer
For a square matrix the answer is yes as the rank of a matrix determines the dimensions of both the row and the column spaces. However for non-square matrices it is not true. For example, take $$A=\begin{bmatrix}1&2&3&4\\0&1&2&3\end{bmatrix}$$ then rows of $A$ are linearly independent but the columns are not.