If the determinant is zero, is it linearly dependent?

Understand the Problem

The question is asking about the relationship between the determinant of a matrix and the linear dependence of the vectors represented by that matrix. Specifically, it inquires whether having a determinant of zero indicates that the vectors are linearly dependent.

More Information

A zero determinant implies that the columns or rows of the matrix are linearly dependent, meaning at least one vector can be written as a linear combination of the others.

Tips

A common mistake is to misinterpret the determinant being zero as implying all rows or columns are zero vectors, which is not necessarily the case.

Sources

• Linear Independence: Definition & Examples - Study.com - study.com
• Why is the determinant zero iff the column vectors are linearly dependent? - Math Stack Exchange - math.stackexchange.com
• Linear Algebra: Linear Independence and Rank - Varsity Tutors - varsitytutors.com