This document provides information about inverse matrices and matrix rank. It defines row echelon form and reduced row echelon form. It explains that the inverse of a matrix A exists if A is invertible/non-singular and can be found using Gaussian elimination or the adjugate method. The rank of a matrix is defined as the maximum number of linearly independent rows, and can be determined by evaluating minors, reduced row echelon form, or normal form. Examples are provided to illustrate inverse and rank calculations.

1. Gandhinagar Institute Of Technology
Subject: Linear Algebra And Vector Calculus
Topic: “Inverse of a matrix and Rank of a
matrix”
Prepared by:
Jaydev Patel(150120119127)
Jaimin Patel(150120119126)
Krunal Patel(150120119128)
Guided by: Prof. Jalpa Patel

2. Contents
• What is a row echolen form?
• Inverse of a matrix
• Rank of a matrix

3. What is a Row echelon form?
What is a Row
reduced echelon form?

4. Conditions for row echelon form:
A matrix is in reduced row echelon form if
• 1. the first nonzero entries of rows are equal to 1
• 2. the first nonzero entries of consecutive rows
appear to the right
• 3. rows of zeros appear at the bottom
• 4. entries above and below leading entries are zero.

5. Examples for RREF:

6. Inverse of a matrix :
• If A is a square matrix & if a matrix B of the
same size can be found such that AB=BA=I,
Then A is said to be invertible & B is called an
inverse of A.
• An invertible matrix is called as a non-singular
matrix.
• A non-invertible matrix is called as a singular
matrix.

7. Using Gaussian elimination to find the inverse:
• To find inverse of B, if it exists, we augment B with the 3
3 identity matrix:
• The strategy is to use Gaussian elimination to reduce [B | I ] to
reduced row echelon form. If B reduces to I , then [B | I ] reduces
to [ I | B‾].
• B inverse(B‾) appears on the right!

8. Reducing the matrix
• Our first step is to get a 1 in the top left of the matrix by using
an elementary row operation:
• Next we use the leading 1 (in red), to eliminate the nonzero entries
below it:

9. • We now move down and across the matrix to
get a leading 1 in the (2; 2) position (in red):
• We can do this by R3  R2
• we use this leading 1 (in red) to eliminate all the nonzero
entries above and below it:

10. • Finally, we move down and across to the (3; 3)
position (in red):
• We make this entry into a leading 1 (in red) and
use it to eliminate the entries above it:

11. We have found B‾
As
• We can check that this matrix is B inverse by verifying
that
B* B‾ = B‾ * B = I .

12. By which we can get the value =

13. By determinant method:
If A is a n*n singular square matrix , then inverse
of the A will be
adjA
Adet
A
)(
11


14. Inverse finding with the help of
determinant:

















ac
bd
Adet
A
thenAdetand
dc
ba
AIf
)(
1
,0)(
1

15. Example :







43
21
A
2)2(3)4(1
43
21



















2
1
2
3
12
13
24
2
11
A

16. Rank of a matrix:--
The positive integer are which is equals to ρ of
rank is said to be the rank of matrix A.
r = ρ (A)
• If it posses the following properties ;
1. There is at least one minor of order are which is
non-zero.
2. Every minor of order (r+1) is zero.
In short the number of non-zero rows in the row
echelon form is known as rank of the matrix.

17. • The rank of a matrix counts the maximum
number of linearly independent rows.
• It is an integer.
• If the matrix is mn, then the rank is an
integer between 0 and min(m,n).

18. • There are three methods for determine the
rank:
1. Rank of a matrix A by evaluating minors :
• Begin with the highest order minor of A, if one
of them is non-zero, then ρ(A)=r. If all the
minors of order ‘r’ are zero, begin with r-1 &
so on till you get a non-zero minor,& that
minor is your RANK.

19. 2. Rank of a matrix A by Row-Echelon form :
• The rank of matrix A in ref=number of non-
zero rows of a matrix.
3. Rank of a matrix by reducing it to Normal
form :
Iᵣ 0 , Iᵣ , [Iᵣ]
0 0 0
• In this method r is obtain as a RANK of A.
[ ] ][

20. Calculation of row-rank via RREF
Row reductions
Row-rank = 2
Row-rank = 2
Because row reductions
do not affect the number
of linearly independent rows

21. Example of Rank :

22. THANK YOU….!!!!