INTRODUCTION to DETERMINANTS

det(A):

For a square matrix A

• If n = 1, det(A) = a₁₁.
• If n = 2, det(A) = a₁₁a₂₂-a₂₁a₁₂.
• For $n \geq 1$, let A_ij be A with row i and col j removed;
  
  $$det(A) = a_{11}det(A_{11})-a_{12}det(A_{12})+\cdots+(-)^{1+n}a_{1n}det(A_{1n}),$$
  
  
  cofactor expansion for det A using first row of A.
• The (i,j)^th cofactor of A is C_ij=(-)^i+jdet(A_ij).
• det(A) can be computed using a cofactor expansion along any row or any column.
• Notation:
  
  $$det(A) = \left\vert \begin{array}{ccc} a_{11}&\dots&a_{1n} \\... ...ts&\vdots \\ a_{n1}&\dots&a_{nn} \\ \end{array} \right\vert.$$
  
  

Easy Determinants

• det(I) = 1.
• det for a triangular A is the product of the diagonal entries.

PROPERTIES OF DETERMINANTS

Row Operations:

• A row interchange for A changes then sign of det(A).
• The scale of a row of A by s changes det(A) to s(det(A)).
• Row replacement operations do not change det(A).
• If $A \sim U$ (upper triangular) and r was the number of row interchanges
  $det(A) = (-)^r det(U) = (-1)^ru_{11}u_{22}\cdots u_{nn}$,
• A square matrix A is invertible iff $det(A) \neq 0$.

Other Properties:

• det(A) = det(A^T).
• det(AB) = det(A)det(B)
  simple consequences:
  • det(A^-1) = (det(A))^-1
  • det(Aⁿ)=(det(A))ⁿ
  • det(sA) = sⁿ det(A)
  • det(AB) = det(BA)

DETERMINANT FORMULAS

Cramer's Rule:

Let $A_i({\bf b})$ denote A with col i replaced by ${\bf b}$.
The solution to $A{\bf x}\ = {\bf b}$ is given by $x_i = det(A_i({\bf b}))/det(A)$, for $i=1,2,\ldots,n$.

A^-1 Formula:

Using adj(A) to denote the adjugate or adjoint of A
(adj(A) is the transposed matrix of cofactors C_ij for A),

$$A^{-1} = \frac{1}{det(A)} adj(A) = \frac{1}{det A}\left[ \be... ...\vdots \\ C_{1n}&C_{2n}&\dots&C_{nn} \\ \end{array} \right].$$

For n=2, $A^{-1} = \frac{1}{a_{11}a_{22}-a_{21}a_{12}} \left[ \begin{array}{cc} a_{22}&-a_{12}\\ -a_{21}&a_{11} \end{array} \right]$.

Areas and Volumes:

• A parallelogram determined by vectors ${\bf v}_1$, ${\bf v}_2$ has area $= \vert det [{\bf v}_1\ {\bf v}_2 ]\vert$.
• A parallelepiped determined by vectors ${\bf v}_1$, ${\bf v}_2$, ${\bf v}_3$ has volume $= \vert det [{\bf v}_1\ {\bf v}_2\ {\bf v}_3]\vert$.