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Matrix Algebra: Why, What and How Course

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This course teaches matrix algebra from beginner level to pro-level. Here we have tried to design this course in such a way that a proper understanding can be developed in students who learn it. The unique feature of this course is that it can make you curious to learn more and enable to use your knowledge for academic purposes as well as for understanding day to day problem where matrix algebra can be used to find solutions.

34 students enrolled

265 total views, 1 views today

This course teaches matrix algebra from beginner level to pro-level. Here we have tried to design this course in such a way that a proper understanding can be developed in students who learn it. The unique feature of this course is that it can make you curious to learn more and enable to use your knowledge for academic purposes as well as for understanding day to day problem where matrix algebra can be used to find solutions.

Matrix algebra is actually a different language which runs parallelly with other mathematical methods. The same mathematical problem you solve using different mathematical formulae becomes altogether a very simple and attractive when solved using rules of matrix algebra. We have also used Excel and R-software to solve problems. The aim was to update students with the current developments taking place in this field.

Main Features

  • Why do we study matrix and what is the meaning of matrix?
  • How mathematical operations are done using matrix algebra.
  • With very colorful boxes without leaving a single step, each concept of matrix algebra has been illustrated.
  • Answers to students questions are also given within a 4-days time.
  • If you want to become an instructor, you can join us.
  • Use of EXCEL and R Software is also taught in this course for matrix algebra.

What is the target audience?

  • From 10th standard onwards, every student can enroll for the course.
  • Teachers who teach it to various class can also enroll.

The course duration depends on the time devoted and learning speed of the student.

Starting Course

1
Why Matrices are studied
10 minutes

There are two foremost reasons why we study matrices:

(1) All real data comes in the form of a matrix whether or not you recognize that as a matrix.

(2) It becomes relatively easy to do mathematical writing when matrix notations are used.

2
Addition of two, three and N Matrices.
5 minutes
For adding two or more matrices, we simply add the corresponding elements of both the matrices. That is, we take the first element, a11, of matrix A and add it to the first element, b11, of matrix B. This is called element-wise addition or 'vector recycling'.


3
Subtraction of two, three and N Matrices.
5 minutes
Subtraction of matrices is also very simple and straight-forward like the addition of matrices. Here, too, the same condition is stipulated that rows and columns of both matrices have to be equal.


4
Rules of Matrix Multiplication
10 minutes
For multiplying two or more matrices you have to understand when multiplying two or more matrices is allowed and when it is not allowed under matrix algebra. 

 

5
Matrix Multiplication: Multiplying two or more matrices-example
10 minutes
The first step in matrix multiplication is to take the first row of the first matrix A (pre-multiplication matrix) and multiply each element of that row one by one by the corresponding elements of the first columns in matrix B.


6
Calculating Covariance and Properties of Covariance
10 minutes
Covariance or Covariance Matrix Calculations
Step-1: Firstly, we have to calculate the mean of each column, i.e, X1, X2, and X3.
 The formula for calculating mean:


7
Calculating Matrix’ mean, variance and standard deviation
5 minutes

Calculating Matrix' mean, variance and standard deviation.

To calculate the mean of a matrix having mxn dimension, calculate the mean of each column because each column is a different variable; then, use these means to find out variances and S.Ds of each variable or column respectively.


8
Matrix Determinant Intro
5 minutes

Matrix Determinant

 

We can calculate the determinant of a matrix only when it is a square matrix.

 

 If it is not a square matrix or it is a rectangular matrix (rows ≠ columns), then we make it a pseudo square matrix.

9
Calculating Determinant of a 2 x 2 Matrix
5 minutes

Calculating the Determinant of a 2x2 matrix:

 

To calculate the determinant of a matrix having 2 rows and 2 columns, we should differentiate the elements of the matrix into diagonal and off-diagonal elements.

10
Calculating the Determinant of a 3 x 3 Matrix

11
Calculating Co-factor matrix and Adjugate or Adjoint of a 2×2 matrix
5 minutes

For a 2x2 matrix, we do not first construct the matrix of minor and then change the sign to make the cofactor matrix, which is later transposed to make an adjoint matrix. Here, the same operations are implemented easily and the adjoint of a 2x2 matrix is found directly making the following adjustments:

12
Cofactor matrix and Adjugate/Adjoint of a 3×3 matrix
Cofactor matrix and Adjugate/Adjoint of a 3x3 matrix
Finding the cofactor and adjugate or adjoint of a 3x3 matrix involves many steps and is a bit lengthy. We use a procedure known as Laplace or cofactor expansion.
Practically speaking, (1) we find out 3x3= 9 determinants from a 3x3 matrix and arrange them into a rectangular box. After this, (2) we put a positive or negative sign in front of each element of that rectangular box, according to rules given below, to make it a cofactor matrix.
If row + column = an even number, then use a (+) sign in front of the number.
If row + column = an odd number, then use a (-) sign in front of the number.
 (3) When the cofactor matrix is calculated, we transpose it to make adjugate or adjoint matrix.


13
Finding the Inverse of a 2×2 matrix
5 minutes
Finding the Inverse of a 2 x 2 matrix
Elements of an inverse matrix are just the quotients obtained solving the fraction having an element of the adjugate/adjoint matrix as a numerator and the determinant of the matrix in the denominator. Since it is a 2 x 2 matrix having 4 elements, then we have to divide each of these 4 elements in the numerator one by one having the determinant of the matrix in the denominator. 
14
Finding the Inverse of a 3 x 3 matrix
5 minutes

Computing the inverse of a 3x3 Matrix

 There are three steps involved in calculating the inverse of a 3x3 matrix as we did in the case of computing the inverse of a 2x2 matrix, but, here, the procedure is not very straight forward as we experienced in a 2x2 matrix and is a bit lengthy in terms of execution of each step.

 (1) First, we calculate the determinant of the matrix.

 (2) We construct the co-factor matrix, which is then transposed to make the adjugate or adjoint matrix.

 (3) Finally, this adjugate matrix is divided by the determinant of this matrix, calculated in the first step.

15
Properties and interpretation of Matrix Inverse
10 minutes
Properties and Interpretation of Matrix Inverse
 
(1) Property of Matrix Inverse
 The inverse is a property of a square matrix.
 In a square matrix, number of rows = number of columns
That is, 2 rows x 2 columns, 3 rows x 3 columns, 4 rows x 4 columns etc

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