Linear algebra, rank and nullity

[vs55]

Homework Statement

Find the rank and nullity of the given matrix:

|-2 2 1 1 -2 |----->(1)
|1 -1 -1 -3 3 |----->(2)
|-1 1 -1 7 5 |----->(3)


The attempt at a solution
i know rank is the number of non-zero rows and nullity is the # of columns minus the rank
matrix:
i took (1)+(2) then (1) x -1 to get:
|1 -1 0 2 -1 |
| 1 -1 -1 -3 3 |
|-1 1 -1 7 5 |

(2)+(3)

|1 -1 0 2 -1 |
|0 0 -2 4 8 |
|-1 1 -1 7 5 |

(3)+(1)
|1 -1 0 2 -1 |
|0 0 -2 4 8 |
|0 0 -1 9 4 |

by this the rank should be 3 right? and the nullity is 5-3 = 2

the correct answer is rank 2 and nullity 3
what am i doing wrong?

 

[CompuChip]

You have probably made some calculation errors.

For example, in your first step, instead of
|1 -1 0 2 -1 |
| 1 -1 -1 -3 3 |
|-1 1 -1 7 5 |

I get

|1 -1 0 2 -1 |
| 1 -1 0 2 -1|
|-1 1 -1 7 5 |

 

[Architeuthis Dux]

Your solution is incorrect because when you added (2) and (3), you did not take into account the negative signs in front of the first and second rows. The correct addition should be:

(2)+(3)
|1 -1 0 2 -1 |
|-1 1 -1 4 8 |
|-1 1 -1 7 5 |

This leads to the same matrix as when you added (1)+(2), and therefore the rank is still 2. However, when you add (3)+(1), you get:

(3)+(1)
|1 -1 0 2 -1 |
|0 0 -2 4 8 |
|0 0 -2 8 4 |

This matrix has a row of zeros, so the rank is now 2. Therefore, the correct answer is rank 2 and nullity 3.