# Rank of matrix problem

find the value for k for which the matrix

A=
| 9 -1 11 |
|-6 5 -16 |
| 3 2 k |

has rank= 2

* the spacing on the matrix doesn't seem to want to stay formatted, but it's a 3X3 with row 1= (9, -1, 11), row 2= (-6, 5, -16) and row 3=(3,2, k)
3. The Attempt at a Solution

- I tried to solve this using row reduction and then solve for k, but I don't think thats right at all. Can someone please explain this problem to me and the technique for solving it?

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rock.freak667
Homework Helper
$$\left( \begin{array}{Ccc} 9 & -1 & 11\\ -6 & 5 & -16 \\ 3 & 2 & k\ \end{array} \right)$$

What you need to is row reduction to Row echelon form, the number of non-zero rows is the rank,2 in this case. So there should be 2 non-zero rows.

$$\left( \begin{array}{Ccc} 9 & -1 & 11\\ -6 & 5 & -16 \\ 3 & 2 & k\ \end{array} \right)$$

What you need to is row reduction to Row echelon form, the number of non-zero rows is the rank,2 in this case. So there should be 2 non-zero rows.
- That is what I tried to do, but then I don't know how to find k. I got this:

row 1= (3, 2, k)
row 2= (0, 9, -16+2k)
row 3= (0, -7, 11-3k)

I don't know how to solve for k at this point, would it just be guess work, or is there a technique to use?

rock.freak667
Homework Helper
Try these operations
$$3R_3-R_1,9R_2+6R1$$

that should give you something better.

Edit: Do you know the easiest method to get to RE form when given a matrix?

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There are a few ways of approaching this, but this is the first that came to my mind.

For it to be rank 2, you should try finding k such that the third row is a linear combination of the first two. So try finding c1 and c2 such that:

c1 R1 + c2 R2 = R3.

The first two components will give you a 2x2 system, and you can then solve for c1 and c2. Then, the third component will give you k.

Hopefully that makes sense and is helpful!

yes, that made it much easier. thanks.

just do determine of matrix zero and find value of k

rock.freak667
Homework Helper
just do determine of matrix zero and find value of k
You did not really have to bump a 2 year old thread.

Heyya Snogger

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A = 9 -1 11
-6 5 -16
3 2 k

= taking 3 common from C1

3 -1 11
-2 5 -16
1 2 k

= R2->R2+2(R3)

3 -1 11
0 9 k-16
1 2 k

=R3->3(R3)-R1

3 -1 11
0 9 k-16
0 7 3k-11

=R2->R2/9

3 -1 11
0 1 k-16/9
0 0 3k-11

=R3->R3+7R1

3 -1 11
0 1 k-16
0 0 3k-66

Since,the rank of the matrix is 2

3k-66=0
3k=66
k=22

Sorry if I m wrong just gave a try...