This document discusses methods for finding the inverse of a matrix. It begins by defining row echelon form (RE form) and reduced row echelon form (RRE form) and the conditions matrices must satisfy to be in these forms. There are two main methods discussed for finding the inverse: Gaussian elimination and using the determinant. The Gaussian elimination method works by augmenting the matrix with the identity matrix and performing row operations to put it in RRE form, where the inverse appears on the right side. For the determinant method, the inverse is equal to the adjugate matrix divided by the determinant. An example calculation demonstrates finding the inverse of a 2x2 matrix using the determinant.

1. GANDHINAGAR INSTITUTE OF
TECHONOLOGY(012)
SUBJECT : Linear Algebra And Vector Calculus
(2110005)
Active Learning Assignment on the topic of
“Inverse of a matrix”
BE Mechanical
Prepared By: Pandya Yash
Guided By :

2. CONTENTS
 Introduction
 RE form & RRE form
 Types of method
 Example

3. RE & RRE form
 What is a Row echelon form ?
 Side figure is the basic formet of RE
form. Here, * is any real number.
 What is a Reduced Row Echelon form ?
 Side figure is the basic formet of RRE
form. Here, * is any real number.
1 * * * *
0 1 * * *
0 0 1 * *
0 0 0 1 *
0 0 0 0 1
1 0 0 0 0
0 1 0 0 0
0 0 1 0 0
0 0 0 1 0
0 0 0 0 1

4. Conditions for RE form
 Every zero row of the matrix occurs a non-zero row.
 The first non-zero number from the left of a non-zero row is 1.
It is known as a leading 1.
 For each non-zero row, the leading 1 appears to the right and
below any leading 1 in the preceding rows.
 For RRE form :
 A matrix said to be in RRE form if each column that contains a
leading 1 in RE form of the matrix has zeros everywhere else in
, that column.

5. Inverse of 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.

6. Types of Method
 There are usually two methods to find the inverse of a
matrix :-
1) Gaussian Elimination mathod &
2) Determinant method.

7. Gaussian Elimination method
 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.
 To find inverse of B, if it exists, we augment B with the 3×3 identity
matrix:

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. Determinant method
 IfA is a n*n singular square matrix , then inverse of
theA will be …
adjA
det(A)
A1

1

14. Example
 If A = and det(A)  0, then
a b
c d
det(A)
A1

1 d -b
-c a

15. Thank You