Singular matrix is defined as 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:

Does linear dependency among columns imply that there is automatically linear dependency among rows of data?

Note: this is a follow-up from an older thread: Singular Matrix and Linear Dependency


I noticed that the question received very few views, so I asked it at the maths community. So here is the answer, in case anyone ever finds it useful.

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.

Further details, an example, and intuition behind this can be found here

  • $\begingroup$ This answer is perhaps too limited to be of much use or insightful. Standard theorems about dimensions of vector spaces provide clear, simple relationships between the numbers of rows, columns, and dimension of the null space of any matrix. A good example to ponder is the $1\times 2$ matrix $\pmatrix{1&1},$ which has an obvious linear dependency among its columns and just as obviously has no linear dependency among its rows. $\endgroup$
    – whuber
    Nov 12 '19 at 15:40

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