Given a square matrix 
, the following are equivalent:
1. 
.
2. The columns of 
are linearly independent.
3. The rows of 
are linearly independent.
4. Range(
)
= 
.
5. Null(
)
= 
.
6. 
has a matrix inverse.
Determinant Theorem
See also
Determinant, Matrix Inverse, Null Space, RangeExplore with Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Determinant Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DeterminantTheorem.html