GATE CSE 2002 | Question: 1.1

Question

The rank of the matrix $\begin{bmatrix} 1 & 1 \\ 0 & 0 \end{bmatrix}$ is

• A. $4$
• B. $2$
• C. $1$
• D. $0$

Comments

Rank of a Matrix is No of Linearly Independent Columns here only $1$ it is that's why rank is $1$

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rank is 1 ,as of my knowledge we find intuitively i.e. number of non zero rows is defined as rank of matrix.

Here, for given matrix rank is 1.

Answer 1

Rank of this matrix is 1 as the determint of 2nd order matrix is 0 and 1st order matrix is non zero so rank is $1$.

Correct Answer: $C$

Comments

$\mathbf{\mid  A_{n\times n} \mid} \neq 0\:\:\textbf{iff}$ rank of the marix $\mathbf{A}$ is $n.$

                                        $\textbf{(OR)}$

$\mathbf{\mid  A_{n\times n} \mid} \neq 0\:\:\Longleftrightarrow$ rank of the marix $\mathbf{A}$ is $n.$

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20 years ago, life was so simple...:D

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but it's C

Answer 2

Eigen Value of a upper triangular matrix = Principal diagonal elements.

The above matrix is a upper triangular matrix, thus the Eigen Value of the above matrix is 1 and 0 (diagnal elemnts).  

Number of non zero Eigen values of a matrix = Rank of that matrix.

Therefore the rank of above matrix=1.

 

Answer 3

Rank Of A Matrix Can be easily calculated by Calculating Order of Submatrix Having Determinant Not Equal to zero in the above submatrix of size 2*2 determinant turns out to be zero hence rank cannot be One but if we see for 1*1 submatrix we have [1] as submatrix whose determinant is not equal to zero hence Rank is 1 .

Answer 4

https://www.youtube.com/watch?v=s6hYAaOehCg

Answer 5

The column rank of a matrix is equal to the number of pivot columns, which is also equal to the number of linearly independent columns.

Hence,there is only one pivot column (i.e., only one linearly independent column), the rank of the matrix is 1.

Answer 6

Very Simple. Just Read this carefully.

Row 1: $[1,1]$ (Contains numbers).

Row 2: $[0,0]$ (All zeros → Ignore it).

Only 1 useful Row.

Rank = 1