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

1. $0$
2. $1$
3. $2$
4. $3$

edited | 1k views

Answer: $C$

Determinant comes out to be $0$. So, rank cannot be $3$. The minor $\begin{bmatrix} 3 & 5 \\[0.3em] 1 & 1 \\[0.3em] \end{bmatrix}\neq0.$ So, rank is $2.$

(OR)

If we do elementary row operations on the given matrix then we get

$\begin{bmatrix}0&0&-3\\9&3&5\\3&1&1\end{bmatrix} \overset{R_2\leftarrow R_2 - 3R_3}{\to} \begin{bmatrix}0&0&-3\\0&0&2\\3&1&1\end{bmatrix}$

$\overset{R_1 \leftarrow R_1 + \frac{3}{2}R_2}{\to} \begin{bmatrix}0&0&0\\0&0&2\\3&1&1\end{bmatrix}\overset{R_1 \leftrightarrow R_3}{\to} \begin{bmatrix}3&1&1\\0&0&2\\0&0&0\end{bmatrix}$

As the number of non zero rows is $2$, the rank of the matrix is also $2.$
by Boss (33.8k points)
edited
+1

if we do elementary row operation then we get :-

as the number of non zero row is 2,then the rank of the matrix is also 2.

–1
0
If any 2x2 minor is != 0 then we can say rank is 2. right?
0
This is a good example of how to calculate rank of a matrix by elementary row operation, not only for GATE but to clear doubt also.
+1
Wat is the operation done in 3rd matrix from left side ??
+2
R1-->3/2R2+ R1
0
Chk column wise, will give rank 2 in single step
+1 vote
$C_1 \rightarrow C_1- 3C_2$

$\begin{bmatrix} 0&0 &-3 \\ 0&3 &5 \\ 0 &1 &1 \end{bmatrix}$

$Rank = 2$
by (23 points)
edited by