Tutor

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.



Tricks: 

You have always to expand by the row or column that contain the highest number of zeros

If a is a triangular matrix then detA is the product of the entries on the main diagonal.

if you didn't get it, please review the video above, it covers all these points.




Homework:     P 182   EX   1-----10    +   18  +  19  +  20  +  26.




Section 4.2:  Properties of Determinants


    Intresting Links:      http://www.wiziq.com/tutorial/8083-properties-of-Determinants

                                                        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 MAT 130 PROPERTIES OD DETERMINANTS.doc
Size : 127 Kb
Type : doc


Row operations for determinants 



Homework:    EX  1----7  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:

\[{\rm{A}} = \left[ {\begin{array}{*{20}c} 1 & 2 & 3 \\ 0 & 4 & 5 \\ 1 & 0 & 6 \\\end{array}} \right]\]
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 det A = 22, we get

.

 


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:



Integrating factor used to solve first order differential equation.



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.