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.
Answer
If the determinant is zero, the set of vectors is linearly dependent.
If the determinant is zero, the set of vectors is linearly dependent.
Answer for screen readers
If the determinant is zero, the set of vectors is 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
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