determinant

what is determinant

• a scalar value
• function of the entries of a square matrix
• considered as scaling factor of a matrix

finding determinant

• det(id matrix A) = 1
• sqr matrix $R^2$: vertical: (a b) (c d) ⇒ $det=ad-bc$
• sqr matrix $R^3$: vertical: (a b c) (d e f) (g h i) ⇒ $det=a(ei-fh)+b(di-fg)+c(dh-ge)$
• sqr matrix $R^4$:

note

• $A\in R^{n,n}$ is singular $⟺detA=0⟺N(A)\ne 0$
• refer to inference in LA
• 

determinant of an orthonormal matrix

• orthonormal n x n matrix, $U^{T}U=U{U}^T=I_n$
• $1=\det I_n=\det U^{T}U=(\det U{)}^2$ → U = +-1