## Rank of a Matrix

In this page rank of a matrix we are going to see how to calculate rank of any matrix with examples.

To find rank of any given matrix first we have to find the echelon form(triangular form)

Procedure to find Echelon form (triangular form)

(i) The first element of every non-zero row is 1.

(ii) The row which is having every element zero should be below the non zero row.

(iii) Number of zeroes in the next non zero row should be more than the number of zeroes in the previous non zero row.

Example 1:

 1 1 -1 3 -2 3 2 -3 4

Solution:

˜

 1 1 -1 3 -2 3 2 -3 4

 R₂ => R₂ - 3R₁ 3        -2         3         3         3        -3        (-)       (-)       (+)      ________________       0         - 5        6      ________________
 R₃ => R₃ - 2R₁ 2        -3         4       2         2        -2      (-)       (-)       (+)       ________________       0         -5       6          ________________

˜

 1 1 -1 0 -5 6 0 -5 6

R₂ => R₂ - 3R₁

R₃ => R₃ - 2R₁

 R₃ => R₃ - R₂ 0        -5         6       0        -5         6      (-)       (+)       (-)      ___________________       0         0        0          __________________

rank of a matrix

˜

 1 1 -1 0 -5 6 0 0 0

R₃ => R₃ - R

Number of non zero rows is 2. So rank of the given matrix = 2.

Example 2:

 4 3 6 25 1 5 7 13 2 9 1 1

Solution:

˜

 4 3 6 25 1 5 7 13 2 9 1 1

R₂ <-> R₁

˜

 1 5 7 13 4 3 6 25 2 9 1 1

 R₂ => R₂ - 4R₁ 4         3         6        25          4         20       28       52        (-)        (-)       (-)       (-)      ____________________________       0        -17      -22      -27      ___________________________ R₃ => R₃ - 2R₁ 2          9         1         1          2         10       14       26        (-)       (-)       (-)       (-)      ___________________________       0        -1       -13      -25      ___________________________

˜

 1 5 7 13 0 -17 -22 -27 0 -1 -13 -25

R₂ => R₂ - 4 R₁

R₃ => R₃ - 2 R₁

 R₃ => 17R₃ - R₂ 0        -17        -221        -425       0        -17         -22           -27        (-)       (+)       (+)             (+)      _________________________________       0         0         -199         -398           ________________________________

˜

 1 5 7 13 0 -17 -22 -27 0 0 -199 -398

R₃ => 17 R₃ - R₂

Number of non zero rows is 2. So rank of the given matrix = 2.Now you can try the following questions to understand this topic much better.

Questions

Solution

1) Find the rank of the following matrix

 2 1 1 5 1 1 1 4 1 -1 2 1

Solution

2) Find the rank of the following matrix

 1 2 1 7 2 -1 2 4 1 1 -2 -1

Solution

3) Find the rank of the following matrix

 2 5 7 52 1 1 1 9 2 1 -1 0

Solution

4) Find the rank of the following matrix

 3 1 -1 2 2 -1 2 6 2 1 -2 -2

Solution

5) Find the rank of the following matrix

 2 -1 3 9 1 1 1 6 1 -1 1 2

rank of a matrix rank of a matrix

Solution