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.
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.
Weisstein, Eric W. "Determinant Theorem." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/DeterminantTheorem.html