PPT: Rank of a Matrix

# PPT: Rank of a Matrix | Engineering Mathematics - Civil Engineering (CE) PDF Download

``` Page 1

RANK OF A MATRIX
Let A be any m  n matrix. Then A consists of n column
vectors a 1, a2 ,....,a, which are m-vectors.
We write rk(A) for the rank of A. Note that we may
compute the rank of any matrix-square or not
DEFINTION:
The rank of A is the maximal number of linearly
independent column vectors in A, i.e. the maximal
number of linearly independent vectors among {a1,
a2,......, a}.
If A = 0, then the rank of A is 0.
Page 2

RANK OF A MATRIX
Let A be any m  n matrix. Then A consists of n column
vectors a 1, a2 ,....,a, which are m-vectors.
We write rk(A) for the rank of A. Note that we may
compute the rank of any matrix-square or not
DEFINTION:
The rank of A is the maximal number of linearly
independent column vectors in A, i.e. the maximal
number of linearly independent vectors among {a1,
a2,......, a}.
If A = 0, then the rank of A is 0.
? ?
Let us see how to compute 2 2 matrix:
:
The rank of a 2 2 matrix A = is given by
() 2   ad bc 0,  since both column  vectors are
independent in this case.
rk(A) = 1 if det(A)
EXAMPLE
ab
rk A if det A
cd
?
?
? ? ? ?
?
?
?
?
?
?
??
00
= 0 but A 0 = ,since both column vectors
00
are not linearly independent, but there is a single column vector that is
linearly independent (i.e. non-zero).
rk(A) = 0 if A = 0
??
?
??
??
?
RANK OF        MATRIX
22 ?
How do we compute rk(A) of m x n matrix?
Page 3

RANK OF A MATRIX
Let A be any m  n matrix. Then A consists of n column
vectors a 1, a2 ,....,a, which are m-vectors.
We write rk(A) for the rank of A. Note that we may
compute the rank of any matrix-square or not
DEFINTION:
The rank of A is the maximal number of linearly
independent column vectors in A, i.e. the maximal
number of linearly independent vectors among {a1,
a2,......, a}.
If A = 0, then the rank of A is 0.
? ?
Let us see how to compute 2 2 matrix:
:
The rank of a 2 2 matrix A = is given by
() 2   ad bc 0,  since both column  vectors are
independent in this case.
rk(A) = 1 if det(A)
EXAMPLE
ab
rk A if det A
cd
?
?
? ? ? ?
?
?
?
?
?
?
??
00
= 0 but A 0 = ,since both column vectors
00
are not linearly independent, but there is a single column vector that is
linearly independent (i.e. non-zero).
rk(A) = 0 if A = 0
??
?
??
??
?
RANK OF        MATRIX
22 ?
How do we compute rk(A) of m x n matrix?
COMPUTING RANK BY VARIOUS
METHODS
1. BY GAUSS ELIMINATION
2. BY DETERMINANTS
3. BY MINORS
4. BY MORMAL FORM
Page 4

RANK OF A MATRIX
Let A be any m  n matrix. Then A consists of n column
vectors a 1, a2 ,....,a, which are m-vectors.
We write rk(A) for the rank of A. Note that we may
compute the rank of any matrix-square or not
DEFINTION:
The rank of A is the maximal number of linearly
independent column vectors in A, i.e. the maximal
number of linearly independent vectors among {a1,
a2,......, a}.
If A = 0, then the rank of A is 0.
? ?
Let us see how to compute 2 2 matrix:
:
The rank of a 2 2 matrix A = is given by
() 2   ad bc 0,  since both column  vectors are
independent in this case.
rk(A) = 1 if det(A)
EXAMPLE
ab
rk A if det A
cd
?
?
? ? ? ?
?
?
?
?
?
?
??
00
= 0 but A 0 = ,since both column vectors
00
are not linearly independent, but there is a single column vector that is
linearly independent (i.e. non-zero).
rk(A) = 0 if A = 0
??
?
??
??
?
RANK OF        MATRIX
22 ?
How do we compute rk(A) of m x n matrix?
COMPUTING RANK BY VARIOUS
METHODS
1. BY GAUSS ELIMINATION
2. BY DETERMINANTS
3. BY MINORS
4. BY MORMAL FORM
POSSIBLE RANKS:
Counting possible number of pivots, we see that rk(A)  m and rk(A) n
for any m  n matrix A.
??
1.  USING GAUSS ELIMINATION
GAUSS ELIMINATION:
Use elementary row operations to reduce A to echelon form. The
rank of A is the number of pivots or leading coefficients in the
echelon form. In fact, the pivot columns (i.e. the columns with pivots
in them) are linearly independent.
Note that it is not necessary to and the reduced echelon form –any
echelon form will do since only the pivots matter.
Page 5

RANK OF A MATRIX
Let A be any m  n matrix. Then A consists of n column
vectors a 1, a2 ,....,a, which are m-vectors.
We write rk(A) for the rank of A. Note that we may
compute the rank of any matrix-square or not
DEFINTION:
The rank of A is the maximal number of linearly
independent column vectors in A, i.e. the maximal
number of linearly independent vectors among {a1,
a2,......, a}.
If A = 0, then the rank of A is 0.
? ?
Let us see how to compute 2 2 matrix:
:
The rank of a 2 2 matrix A = is given by
() 2   ad bc 0,  since both column  vectors are
independent in this case.
rk(A) = 1 if det(A)
EXAMPLE
ab
rk A if det A
cd
?
?
? ? ? ?
?
?
?
?
?
?
??
00
= 0 but A 0 = ,since both column vectors
00
are not linearly independent, but there is a single column vector that is
linearly independent (i.e. non-zero).
rk(A) = 0 if A = 0
??
?
??
??
?
RANK OF        MATRIX
22 ?
How do we compute rk(A) of m x n matrix?
COMPUTING RANK BY VARIOUS
METHODS
1. BY GAUSS ELIMINATION
2. BY DETERMINANTS
3. BY MINORS
4. BY MORMAL FORM
POSSIBLE RANKS:
Counting possible number of pivots, we see that rk(A)  m and rk(A) n
for any m  n matrix A.
??
1.  USING GAUSS ELIMINATION
GAUSS ELIMINATION:
Use elementary row operations to reduce A to echelon form. The
rank of A is the number of pivots or leading coefficients in the
echelon form. In fact, the pivot columns (i.e. the columns with pivots
in them) are linearly independent.
Note that it is not necessary to and the reduced echelon form –any
echelon form will do since only the pivots matter.
EXAMPLE
Gauss elimination:
* Find the rank of a matrix
1 0 2 1
A= 0 2 4 2
0 2 2 1
??
??
??
??
??
```

## Engineering Mathematics

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## FAQs on PPT: Rank of a Matrix - Engineering Mathematics - Civil Engineering (CE)

 1. What is the rank of a matrix?
Ans. The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It represents the dimension of the vector space spanned by the rows or columns of the matrix.
 2. How is the rank of a matrix calculated?
Ans. The rank of a matrix can be calculated using various methods, such as row reduction or determinants. Row reduction involves performing elementary row operations to transform the matrix into its reduced row echelon form, and the number of non-zero rows in the reduced form corresponds to the rank. Determinants can also be used to determine the rank by examining the non-zero determinants of certain submatrices.
 3. What does a matrix with full rank mean?
Ans. A matrix is said to have full rank if its rank is equal to the maximum possible rank, which is the minimum of the number of rows and columns in the matrix. In other words, a matrix with full rank has linearly independent rows and columns, and its rows and columns span the entire vector space.
 4. Can a matrix have a rank greater than its dimensions?
Ans. No, a matrix cannot have a rank greater than its dimensions. The rank of a matrix is always less than or equal to the minimum of the number of rows and columns in the matrix. If the rank is equal to the dimensions, then the matrix has full rank.
 5. How is the rank of a matrix useful in applications?
Ans. The rank of a matrix plays a crucial role in many areas of mathematics and engineering. It helps in determining the solutions to systems of linear equations, finding the inverse of a matrix, analyzing the linear dependence or independence of vectors, and understanding the properties of linear transformations. Additionally, the rank-nullity theorem relates the rank of a matrix to its nullity, providing insights into the dimension of the null space.

## Engineering Mathematics

65 videos|121 docs|94 tests

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