Welcome to MAT 130. Linear Algebra.
Determinant of a Matrix:
2 by 2 matrix:
3 by 3 Matrix:
NxN Matrix:
Before proceeding to the calculation of the determinant of an NxN matrix, you have to learn about some concepts such as the Minors and the Cofactors. Please have a look at the following link:
http://www.utdallas.edu/dept/abp/PDF_Files/LinearAlgebra_Folder/Cofactors.pdf
Do not pay a great attention to the section where we are talking about the adjoint and the Inverse, we will discuss this stuff later on.
Please have a look at the following Video:
PS/ The Cofactor Expansion provides a method for taking the determinant of a matrix which is fairly useful when computations are being done manually.
Section 4.2: Properties of Determinants
Intresting Links: http://www.wiziq.com/tutorial/8083propertiesofDeterminants
http://tutorial.math.lamar.edu/Classes/LinAlg/DeterminantProperties.aspx
You can also have a look at the word file bellow; It's pertty much intesting and well presented. Hope you get full advantage of it.
MAT 130 PROPERTIES OD DETERMINANTS.doc Size : 127 Kb Type : doc 

Row operations for determinants
Homework: EX 17 Page; 192
Section 4.3: Cramer's Rule; Formula for the inverse of A ; Applications of determinants
Adjoint
The matrix formed by taking the transpose of the cofactor matrix of a given original matrix. The adjoint of matrix A is often written adj A.
Note: This is in fact only one type of adjoint. More generally, an adjoint of a matrix is any mapping of a matrix which possesses certain properties. Consult a book on linear algebra for more information.
Example:  Find the adjoint of the following matrix: 
Solution:  First find the cofactor of each element. As a result the cofactor matrix of A is Finally the adjoint of A is the transpose of the cofactor matrix: 
.
Adjoint method for finding the inverse: A^{1} = (adjoint of A) or A^{1} = (cofactor matrix of A)^{T} 
Example: The following steps result in A^{1} for . The cofactor matrix for A is , so the adjoint is . Since . 
Solving a 3 X 3 Linear System using Cramer's Rule:
Diagonalization : " Eigenvalues and Eigenvectors "
Please Study these Links:
http://sosmath.com/matrix/eigen2/eigen2.html
Differential Equations:
Introduction to differential equations:
Separable differential equations:
Second example:
If You prefer handouts, please try the following links:
http://wwwmath.mit.edu/~djk/18_01/chapter15/contents.html
Lagrange Multipliers
Lagrange Multiplier.
A second lecture
A third well structured lecture.
In this thrid video, the lecturer focused only on explaining the lagrange methode as a whole and independent concept. He did not talk about maximum or minimum.
if you want to learn more about maximum and minimum, please have a look at the two videos above.
Real world example. #1.
Real world example. #2.