LINEAR ALGEBRA AND VECTOR CALCULUS

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

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

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

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.

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.

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!

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:

• 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:

• 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:

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

By which we can get the value =

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

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

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

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.

• 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).

• 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.

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.
[ ] ][

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

LINEAR ALGEBRA AND VECTOR CALCULUS

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LINEAR ALGEBRA AND VECTOR CALCULUS