GATE 1994 Example on Rank of Matrix

The rank of matric \begin{bmatrix} 0 & 0 & -3 \\ 9 & 3 & 5 \\ 3 & 1 & 1\\ \end{bmatrix} is :

a) 0

b) 1

c) 2

d) 3

Answer

Things you need to know

The rank of a matrix is the maximum number of independent rows (or, the maximum number of independent columns).

 

Now,

if we observe, then the column vector \begin{bmatrix} 0\\ 9 \\ 3 \end{bmatrix}is a multiple of  \begin{bmatrix} 0 \\ 3 \\ 1 \end{bmatrix}

Therefore, The total number of linearly independent column vectors = 2

Hence, Rank = 2 .

 

4Comments
Bharath Sathuri @bharathsathuri
29 Oct 2019 09:44 am
where is column vector [0 9 1] in the given matrix?
Shamshad Hussain saifi @shamshadhussain
30 Oct 2019 04:18 pm

@bharath Sathuri :there is a small correction here instead of this column vector [0 9 1] the correct column vector is [0 9 3]. We have corrected that column vector in the above solution also you can  see. And this question  may be solved by observing the number of independent rows.The number of independent rows is two when you apply some elementary transformation  therefore rank =2

 

SHIVAM KUMAR @shivamkumar12
30 Oct 2019 07:34 pm
it's not [0 9 1] . it's [0 9 3]
Shamshad Hussain saifi @shamshadhussain
31 Oct 2019 11:44 am
@shivam KUMAR :yes ,you are correct .It will be [0 9 3]