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
??
??
??
??
??