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33 votes
33 votes
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 $2.$
edited by
3 votes
3 votes
$C_1 \rightarrow C_1- 3C_2$

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

$Rank = 2$
edited by
1 votes
1 votes

Answer: 2

Rank can be defined as:

$\equiv $ Linearly Independent Rows

$\equiv $ Linearly Independent Columns

$\equiv $ Pivot elements (count) in Echelon form of the Matrix

$\equiv $ Non-zero rows of Echelon form of the Matrix

Given matrix:

$\begin{bmatrix}
0 & 0 & -3\\ 
9 & 3 & 5\\ 
3 & 1& 1 
\end{bmatrix}$

Let’s convert it to Echelon Form.

$\equiv \begin{bmatrix}
0 & 0 & -3\\ 
9 & 3 & 5\\ 
3 & 1& 1 
\end{bmatrix} \xrightarrow{R_1 \leftrightarrow R_3} \begin{bmatrix}
3 & 1 & 1\\ 
9 & 3 & 5\\ 
0 & 0 & -3 
\end{bmatrix}$

$\equiv \begin{bmatrix}
3 & 1 & 1\\ 
9 & 3 & 5\\ 
0 & 0 & -3 
\end{bmatrix} \xrightarrow[]{R_2 \rightarrow R_2 - 3R_1} \begin{bmatrix}
3 & 1 & 1\\ 
0 & 0 & 2\\ 
0 & 0 & -3 
\end{bmatrix}$

$\equiv \begin{bmatrix}
3 & 1 & 1\\ 
0 & 0 & 2\\ 
0 & 0 & -3 
\end{bmatrix} \xrightarrow[]{R_3 → 3R_2 + 2R_3} \begin{bmatrix}
3 & 1 & 1\\ 
0 & 0 & 2\\ 
0 & 0 & 0 
\end{bmatrix}$

In the resultant matrix, we have two Pivot elements as marked with $\color{Red} red$:

$\begin{bmatrix}
\color{Red} 3& 1 & 1\\ 
0 & 0 & \color{Red} 2\\ 
0 & 0 & 0 
\end{bmatrix}$

Therefore, rank is 2.

0 votes
0 votes

Do row operation:

R₂→R₂-3R₃

R₁→R₁+(3/2)R₂

Then you a get a matrix whose first row is [0 0 0] and other two row are LI

. Hence rank 2...

 

 


 

 


Answer:

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