II B.Com

1. Rank of a Matrix
Dr. R. MUTHUKRISHNAVENI
SAIVA BHANU KSHATRIYA COLLEGE
ARUPPUKOTTAI

2. Matrix
• Matrices are one of the most commonly used tools in many fields such as
Economics, Commerce and Industry. We have already studied the basic
properties of matrices. In this chapter we will study about the elementary
transformations to develop new methods for various applications of
matrices.

3. Concept of Rank of a matrix
• Enables us to bring together a number of loose ends matrix theory and to
develop some basic ideas in more details. It also plays an important role in
the application of matrices to linear problems. It helps us to find the
consistency of a system of simultaneous linear equations (Next Presentation)
• With each matrix, we can associate a non-negative integer called its rank.
• The maximum number of linearly independent rows in a matrix A is called
the row rank of A, and the maximum number of linearly independent
columns in A is called the column rank of A.

4. Rank of A matrix
• There are two methods
• Determinant Methods
• Elementary method(either row or column)/Gauss Elimination method

5. Determinant Methods
• In a non-zero matrix A of order (m x n), if at least one minor of order r is
not zero and every minor (r + 1) is zero, then r is said to be the rank of the
matrix A and is denoted by p(A)
• (i) p(A) ≥0
• (ii) If A is a matrix of order m x n , then p(A) ≤ minimum of {m,n}
• (iii) The rank of a zero matrix is ‘0’
• (iv) The rank of a non- singular matrix of order n x n is ‘n’
• (v) The rank of a singular matrix of order n x n is n-1, if minor of any one row is not equal
to zero

6. Square Matrices (2x2 matrix)
• Find the rank of the matrix A=
1 5
3 9
• Solution A =
1 5
3 9
• It is a 2 x 2 matrix so, the rank of Matrix A (p(a)) ≥ 2
• Determinant of A (|A|) = 9 – 15 = -6 ≠ 0
• Therefore Rank of Matrix A = 2

7. Square Matrices (3x3 matrix)
• Find the rank matrix A =
3 4 2
5 2 3
4 0 5
• A =
3 4 2
5 2 3
4 0 5
• |A| = 3(10-0) – 4(25-12)+2(0-8) =30-52-16 = -38 ≠ 0
• p(A) = 3
• The rank of Matrix A = 3

8. Square Matrices (3x3 matrix)
• Find the rank matrix A =
1 2 3
4 5 6
7 8 9
• A =
1 2 3
4 5 6
7 8 9
• |A| = 1(45-48) -2(36-42)+3(32-35) = -3+12-9 = 0
• Minor of any one row, we take 1st row 1st Column
5 6
8 9
= 45-48 = -3≠ 0
• p(A) = n-1 = 3-1 =2
• The rank of Matrix A = 2