Determinant

Imagine the area of parallelogram created by the basis of a standard vector space, like . Now apply a linear transformation to that vector space. The new area of the new parallelogram has been scaled by a factor of the determinant. is the parallelopiped.

You can also just think of it as the area of the parallelogram spanned by the columns of a matrix

: parallelopiped and volume

(assume n by n matrix because we only know how to find determinants for square matrices)

You can also get the area of S by using the determinant of the matrix created by the vectors that span S, i.e. because you are shifting the standard basis vectors into the vector space dictated by S

Determinant Laws

• det(A) = 0 A is singular
  • det(A) 0 A is invertible
• det(Triangular) = product of diagonals
• det A = det
• det(AB) = det A · det B

Determinant Post Row Operations

if A square:

• if adding rows to rows on A to get B then
• if swapping rows in A to get B then
• if scaling one row of A by k, then = Exactly the same for columns