Equality of Vectors:  

Definition 1:

Vectors  V=(a,b,c,...,z)  and    W=(1,2,3,4,5,...,n) are said to be equivalent if   a=1 ;  b=2 ; c=3 ;  ... 

We indicate the equality by writing  V=W

Definition 2:

V+W= (a+1, b+2, ...,Z+n) 

KV= (Ka, Kb, Kc, ..., Kn)



As far as we know there are two notations for vectors: 

The comme delimited form: V=(a,b,c,...,z) also called the row vector form

The column vector form

Will start this cours by an Introduction to matrices: 

Elementary row operations:

  1. Multiply an equation through by a nonzero constant
  2. Interchange two equations
  3. Add a multiple of one equation to another
The Importance and significant of these elementary rwo operations will be highlighted in further sections when we will be talking about elemnetary matrices and solving systems using the inverse. That's why we advise you to study them deeply and to remember the difference between them.

Echelon Forms:

 Before getting started with Jordan and Gaussian you have first to learn some properties about echelon forms: 

                        A matrix is in Row Echelon Form if:

  1. The zero rows are grouped together at the bottom of the matrix.
  2. In any two successive rows, the leading entry of the second one should be on the right of the leading entry of the first one.
  3. In a non zero row the leading entry is always one.

                       A matrix is in Reduced Row Echelon Form if:

  1. It is in Row Echelon form.
  2. Eache column that contain a leading entry 1 has zeros everywhere.

                    Every matrix has a unique reduced row echelon form.
                    Every matrix has an infinite number of row echelon forms.

Augmented Matrices and Jordan-Gausse and Gaussian elimination:

Solving Linear Systems by Row Reduction: 

Matrix Operations:  

The Trace:

If A is a square matrix, then the trace of A, denoted by tr(A), is defined to be the sum of the entries on the main diagonal of A.

Zero matrix  VS Identity matrix 

  • A is a zero matrix if and only if all its entries are equal to ZERO
  • A square matrix with 1's on the main diagonal and zeroq elswhere is called an identity matrix.

Inverse of Matrix:  Theory  

Inverse of a 2 by 2 Matrix:

The inverse of a matrix using row operations. 

here it's a 3 by 3 but it workes for all the sizes 

Properties of Diagonal Matrix and Transpose:  


Elementary matrices; A method for finding the inverse of the matrix A 


Solving  Linear Systems by matrix inversion:




Spaning and Basis 


Some practice: 

This is it.The End 

This is almost all about Mat 120, we hope that you enjoyed our company and that you took full advantage of what we presented to you.

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