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15 votes
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The rank of the matrix given below is:

$$\begin{bmatrix} 1 &4 &8 &7\\ 0 &0& 3 &0\\ 4 &2& 3 &1\\ 3 &12 &24 &21 \end{bmatrix}$$

  1. $3$
  2. $1$
  3. $2$
  4. $4$
in Linear Algebra
edited by
2.3k views

3 Answers

25 votes
 
Best answer

$\begin{bmatrix} 1 &4 &8 &7\\ 0 &0& 3 &0\\ 4 &2& 3 &1\\ 3 &12 &24 &21 \end{bmatrix} = 3\begin{bmatrix} 1 &4 &8 &7\\ 0 &0& 3 &0\\ 4 &2& 3 &1\\ 1 &4 &8 &7 \end{bmatrix} $

$R_1$ and $R_4$ are the same and hence we can remove $R_4$ making the rank surely less than $4$.

$\text{Taking 3 out from $R_2$} \implies  9\begin{bmatrix} 1 &4 &8 &7\\ 0 &0& 1 &0\\ 4 &2& 3 &1 \end{bmatrix} $

${R_{1} \leftarrow R_{1}-8R_{2} \atop{R_{3} \leftarrow R_{3}-3R_{2}} } \implies  9\begin{bmatrix} 1 &4 &0 &7\\ 0 &0& 1 &0\\ 4 &2& 0 &1 \end{bmatrix} $

None of the rows are linearly independent (we cannot make any of them all $0's$. 

So, Rank will be $\textbf{3}$.

(A) is correct option!


edited by
4

see this..

1
If you have a look at your solution again, you will find that you've jotted down the question wrong.The element at (2,4) is 21( not 2) which would make the first and second rows linearly dependent. Hence rank can't be 4.

It is 3.
2
If we take 2 then it would be rank =4 and if we consider 21 then rank would  be 3
0
Perfect :)
2 votes
Correct Question -->

The rank of the matrix given below is:

    1    4    8     7
    0    0    3     0
    4    2    3     1
    3   12  24    21

 

 

 

Since R4=3R1 Then Rank != 4

now try for rank of 3

                1  4  8          
                0  0  3   = -3 *   1   4    = -3 * -14 =52
                4  2  3               4   2
  
      here 52 != 0
    So, Rank of the given matrix is = 3
1 vote
Answer D. 4
To calculate matrix rank transform matrix to upper triangular form using elementary row operations.
  A1 A2 A3 A4
1 1 4 8 7
2 0 0 3 0
3 4 2 3 1
4 3 12 24 2

Multiply the 1st row by 4.  R1->R1×4

  A1 A2 A3 A4
1 4 16 32 28
2 0 0 3 0
3 4 2 3 1
4 3 12 24 2

Subtract the 1st row from the 3rd row and restore it       R3->R3-R1

  A1 A2 A3 A4
1 1 4 8 7
2 0 0 3 0
3 0 -14 -29 -27
4 3 12 24 2

Multiply the 1st row by 3.  R1-->R1×3

  A1 A2 A3 A4
1 3 12 24 21
2 0 0 3 0
3 0 -14 -29 -27
4 3 12 24 2

Subtract the 1st row from the 4th row and restore it.    R4-->R4-R1

  A1 A2 A3 A4
1 1 4 8 7
2 0 0 3 0
3 0 -14 -29 -27
4 0 0 0 -19

Swap the 2nd and the 3rd rows R2<->R3

  A1 A2 A3 A4
1 1 4 8 7
2 0 -14 -29 -27
3 0 0 3 0
4 0 0 0 -19

Calculate the number of linearly independent rows

  A1 A2 A3 A4
1 1 4 8 7
2 0 -14 -29 -27
3 0 0 3 0
4 0 0 0

-19

 
Answer:

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