Find the rank and nullity of the matrix
[[1, - 2, 2, 3, 6]
[0, - 1, - 3, 1, 1]
[- 2, 4, - 3, - 6, 11]]
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Answer:
A=
⎣
⎢
⎢
⎡
1
4
3
2
5
4
3
6
5
⎦
⎥
⎥
⎤
Here, ∣A∣=1−4+3=0
So, rank of matrix <3.
Now, we will reduce into Echelon form,
R
2
→R
2
−4R
1
⎣
⎢
⎢
⎡
1
0
3
2
−3
4
3
−6
5
⎦
⎥
⎥
⎤
R
3
→R
3
−3R
1
⎣
⎢
⎢
⎡
1
0
0
2
−3
−2
3
−6
−4
⎦
⎥
⎥
⎤
R
3
→R
3
−
3
2
R
2
⎣
⎢
⎢
⎡
1
0
0
2
−3
0
3
−6
0
⎦
⎥
⎥
⎤
R
2
→−
3
1
R
2
⎣
⎢
⎢
⎡
1
0
0
2
1
0
3
2
0
⎦
⎥
⎥
⎤
Here, there are 2 non-zero rows in the Echelon form of A.
Hence, rank =2
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