Rank of Matrix Problem: Finding k for Rank=2 | Explanation & Solution

In summary, to find the value of k for which the matrix A has rank 2, you need to use row reduction to get it into row echelon form. Then, you can set up a system of equations to find the values of c1 and c2 that make the third row a linear combination of the first two. Finally, the third component will give you the value of k. In this conversation, the value of k is found to be 22.
  • #1
snoggerT
186
0
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)

The Attempt at a Solution



- I tried to solve this using row reduction and then solve for k, but I don't think that's right at all. Can someone please explain this problem to me and the technique for solving it?
 
Last edited:
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  • #2
[tex]\left(
\begin{array}{Ccc}
9 & -1 & 11\\
-6 & 5 & -16 \\
3 & 2 & k\
\end{array}
\right)[/tex]

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.
 
  • #3
rock.freak667 said:
[tex]\left(
\begin{array}{Ccc}
9 & -1 & 11\\
-6 & 5 & -16 \\
3 & 2 & k\
\end{array}
\right)[/tex]

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?
 
  • #4
Try these operations
[tex]3R_3-R_1,9R_2+6R1[/tex]that should give you something better.

Edit: Do you know the easiest method to get to RE form when given a matrix?
 
Last edited:
  • #5
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!
 
  • #6
yes, that made it much easier. thanks.
 
  • #7
just do determine of matrix zero and find value of k
 
  • #8
sanjeevece said:
just do determine of matrix zero and find value of k

You did not really have to bump a 2 year old thread.
 
  • #9
Heyya Snogger
 
Last edited:
  • #10
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...
 

What is a rank of a matrix?

The rank of a matrix is the maximum number of linearly independent rows or columns in the matrix. It is also known as the dimension of the vector space spanned by its rows or columns.

How do you find the rank of a matrix?

The rank of a matrix can be found by performing row reduction operations on the matrix until it is in reduced row echelon form. The number of non-zero rows in the reduced matrix is the rank of the original matrix.

What does a rank of 0, 1, or n mean for a matrix?

A rank of 0 means that all rows and columns in the matrix are linearly dependent, meaning they can be expressed as a combination of other rows or columns. A rank of 1 means that all rows or columns in the matrix are multiples of each other. A rank of n, where n is the number of rows or columns in the matrix, means that all rows and columns are linearly independent.

How does the rank of a matrix affect its invertibility?

A square matrix is invertible if and only if its rank is equal to its number of rows (or columns). If the rank is less than the number of rows, the matrix is singular and cannot be inverted.

Can two matrices with different dimensions have the same rank?

Yes, two matrices with different dimensions can have the same rank. This can happen if the matrices have the same number of linearly independent rows or columns, even though their total number of rows or columns may be different.

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