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:
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Solution:
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R₂ => R₂ - 3R₁ |
3 -2 3 3 3 -3 (-) (-) (+) ________________ 0 - 5 6 ________________ |
R₃ => R₃ - 2R₁ |
2 -3 4 2 2 -2 (-) (-) (+) ________________ 0 -5 6 ________________ |
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R₂ => R₂ - 3R₁ R₃ => R₃ - 2R₁ |
R₃ => R₃ - R₂ |
0 -5 6 0 -5 6 (-) (+) (-) ___________________ 0 0 0 __________________ |
rank of a matrix |
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R₃ => R₃ - R₂ |
Number of non zero rows is 2. So rank of the given matrix = 2.
Example 2:
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Solution:
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R₂ <-> R₁ |
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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 ___________________________ |
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R₂ => R₂ - 4 R₁ R₃ => R₃ - 2 R₁ |
R₃ => 17R₃ - R₂ |
0 -17 -221 -425 0 -17 -22 -27 (-) (+) (+) (+) _________________________________ 0 0 -199 -398 ________________________________ |
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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 |
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2) Find the rank of the following matrix |
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3) Find the rank of the following matrix |
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4) Find the rank of the following matrix |
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5) Find the rank of the following matrix |
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rank of a matrix rank of a matrix |