Determinants

Definition: The determinant is a function from the set of square matrices to the set of real numbers with the following properties:

det(I)=1, where I is the identity matrix
det(A^T)=det(A) where (A^T) is the transpose of the matrix A
det(A)=0 if and only if the rows of A are linearly dependent.
det(A \cdot B)=det(A) \cdot det(B)

Regulations: The determinant of a matrix A is denoted det(A) or with straight lines:

\det \left( {\matrix{ a & b \cr c & d } } \right) = \left| {\,\,\matrix{ a & b \cr c & d } \,\,} \right|\,\, and \det \left( {\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr } } \right) = \left| {\,\,\matrix{ a & b & c \cr d & e & f \cr g & h & i \cr } \,\,} \right|\,\,

For 2x2 matrices and for 3x3 matrics the determinant is calculated as follows:

2х2 matrix	3х3 matrix
\left\| {\,\,\matrix{ a & b \cr c & d } \,\,} \right\| = ad-bc

Example 1: Calculate: \det \left( {\matrix{ 2 & 1 \cr 4 & 3 } } \right)?

Solution: \det \left( {\matrix{ 2 & 1 \cr 4 & 3 } } \right) = \left| {\,\,\matrix{ 2 & 1 \cr 4 & 3 } \,\,} \right|=2 \cdot 3-4 \cdot 1=6-4=2\,\,

Answer: \det \left( {\matrix{ 2 & 1 \cr 4 & 3 } } \right)=2.

Example 2: Calculate: \eqalign{& \det \left( {\matrix{ 1 & 0 & { - 1} \cr { - 1} & 2 & 0 \cr 3 & 1 & 2 \cr } } \right) } ?

Solution:

\eqalign{ & \det \left( {\matrix{ 1 & 0 & { - 1} \cr { - 1} & 2 & 0 \cr 3 & 1 & 2 \cr } } \right) = \left| {\,\,\matrix{ 1 & 0 & { - 1} \cr { - 1} & 2 & 0 \cr 3 & 1 & 2 \cr } \,\,} \right|\,\,\matrix{ 1 \cr { - 1} \cr 3 \cr } \,\,\,\,\matrix{ 0 \cr 2 \cr 1 \cr } \,\, \cr & \,\,\,\,\,\,\,\, = 1 \cdot 2 \cdot 2 + 0 \cdot 0 \cdot 3 + ( - 1) \cdot ( - 1) \cdot 1 - 3 \cdot 2 \cdot ( -1) - 1 \cdot 0 \cdot 1 - 2 \cdot ( - 1) \cdot 0 \cr & \,\,\,\,\,\,\,\, = 4 + 0 + 1 + 6 - 0 - 0 = 11 \cr }

Answer:

\eqalign{ & \det \left( {\matrix{ 1 & 0 & { - 1} \cr { - 1} & 2 & 0 \cr 3 & 1 & 2 \cr } } \right)}=11.