Definition: Determinant
- The geometric interpretation is this is the area of the parallelogram formed by the two vectors
- This expands to higher dimensions as well
- For the determinant of a 3 by 3 matrix, it would be the volume of the parallel piped in 3 dimensions
Definition: Determinant of a 3 by 3 matrix
- The reason that determinants are useful is because they give us the constant that any area on the plane is increased or decreased by following a linear transformation
- This can give use information about the nature of the transformation
- For some matrix
, if , then the following are true: - The matrix is invertible (non-singular)
- Its columns/rows are linearly independent
- The associated linear system
has a unique solution - The linear transformation preserves volume (meaning that it does not collapse the space to a lower dimension)
- The matrix has full rank
- In the case that
- The matrix is non-invertible
- Its columns/rows are linearly dependent
- The associated linear system has no solution or infinitely many solutions
- The linear transformation collapses the space to a lower dimension
- The matrix is rank-deficient
Property:
Property:
\det (XY)=\det X\det Y