RANK OF THE MATRIX USING ECHELON FORM

RANK OF THE MATRIX:
DEFINITION OF RANK : The rank of a matrix A is said to of rank 'r' ,

• if all of its (r+1) order minor A is zero
• there exists atleast one rth order minor of A which is not zero.

Rank of A is denoted by ρ(A).

NOTE:

1. Rank of a matrix is unique
2. Every matrix will have a rank
3. Rank of identity matrix of order nxn is 'n'
4. The rank of a matrix is >= r, if there is atleast one minor of rth order which is not equal to zero.
5. The rank of a matrix doesnt change if elementary row operations are performed.

ECHELON FORM OF MATRIX:

A matrix is said to be in echelon form if it has the following property

1. Zero rows are below the non zero rows
2. The number of zeros before the first nonzero element in a row is less than the no. of zeros in the next row.

The echelon form of the matrix can be obtained by performing ROW OPERATION only.

RANK OF THE MATRIX USING ECHELON FORM:

THE RANK OF THE MATRIX IS DETERMINED BY THE NO. OF NON ZERO ROWS IN THE ECHELON FORM OF THE MATRIX.

EXAMPLE FOR FINDING THE RANK

Find the rank of the matrix

1 -2 0 1

2 -1 1 0

3 -3 1 1

-1 -1 -1 1

Soln: R2 -> R2 - 2R1, R3 -> R3-3R1 and R4 -> R4+R1, we get

1 -2 0 1

0 3 1 -2

0 3 1 -2

0 -3 -1 2

R3-> R3-R2 and R4-> R4+R2

1 -2 0 1

0 3 1 -2

0 0 0 0

0 0 0 0

This matrix is in Echelon form. In the above matrix number of non-zero rows is 2.

Hence rank (A) = 2

IMPORTANT QUESTIONS :

Define the rank of the matrix and find the rank of the following matrix

2 1 3 5

4 2 1 3

8 4 7 13

8 4 -3 -1

Soln: the row operations to be followed are--

R2-> R2-2R1 ; R3->R3-4R1 ; R4->R4-4R1

Then R3-> R3- R2 ; R4-> R4-3R2

Ans: Rank of A = 2

SOME TYPES OF MATRICES--DEFINITION AND PROPERTIES

Symmetric Matrix
A square matrix A= [aij] is symmetric if aij = aji - for every i and j
A= A^T
Skew Symmetric Matrix
A square matrix A=[aij ] is skew symmetric if aij = -aji . for every i and j
A= -A^T
Orthogonal Matrix
A square matrix A is orthogonal if AA^T = A^TA = I

Important Theorems And Properties

1. Every sq. matrix can be expressed as the sum of a symmetric and skew symmetric matrix
2. Inverse of a non-singular symmetric matrix A is symmetric
3. If A is a symmetric matrix then adj(A) is also symmetric
4. If A,B are orthogonal matrices then AB and BA are orthogonal
5. Inverse of orthogonal matrix is orthogonal and its transpose is also orthogonal
6. A,B are invertible matrices then (AB)^-1 = (B)^-1(A)^-1

IMPORTANT QUESTIONS:

Q1.show that 1/2 mat (A) is orthogonal where mat(A) is
-1 1 1 1
1-1 1 1
1 1-1 1
1 1 1-1
soln : Find A^T
By Definition AA^T = I
Solve for LHS (the ans. is a 4x4 unit matrix)
Hence LHS = RHS


Q2. show that A³ = A^-1 where A is
3 -3 4
2 -3 4
0 -1 1

soln : solve A² and assume as B
then solve AB
to solve A^-1
A^-1 = adj (A) / A
to solve adj A
the cofactor for every aij is to be taken
cofactor, is (-1)i+j Mij where Mij is minor of every aij

Unit1 - Chapter1 - MATRICES AND LINEAR SYSTEM OF EQUATIONS

THE IMPORTANT TOPICS IN THE UNIT ARE :

Introduction

Elementary Row Operations

Rank

Echelon Form

Normal Form

Solution Of Linear Systems

LU Decomposition

Solution Of Tridiagonal Systems

Solution Of Linear Systems through Direct Methods

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