# Matrices with infinitely many solutions

• Sep 16th 2007, 03:37 PM
Thomas
Matrices with infinitely many solutions
What does this look like?

I'm given a matrix which has a variable in it, and the question asks to determine the value variable such that the augmented matrix of the linear system has infinitely many solutions.
• Sep 16th 2007, 03:55 PM
Plato
Gee, I think we need to see the exact question.
• Sep 16th 2007, 08:44 PM
Thomas
Determine the value of $\displaystyle h$ such that the matrix is the augmented matrix of a linear system with infinitely many solutions.

$\displaystyle \begin{array}{c}\ \\ \\ \end{array}\;\begin{vmatrix}\;7 & -7 & 3\;\\\;14 & h & 6\end{vmatrix}$

That's the exact question. :)
• Sep 16th 2007, 09:13 PM
Jhevon
Quote:

Originally Posted by Thomas
Determine the value of $\displaystyle h$ such that the matrix is the augmented matrix of a linear system with infinitely many solutions.

$\displaystyle \begin{array}{c}\ \\ \\ \end{array}\;\begin{vmatrix}\;7 & -7 & 3\;\\\;14 & h & 6\end{vmatrix}$

That's the exact question. :)

ok. what do we need to have happen in order for there to be infinitely many solutions?
• Sep 17th 2007, 04:25 AM
Thomas
That's what I'm asking... I'm not sure.

I do notice that the second equation is double the first (except for $\displaystyle h$.)
• Sep 17th 2007, 05:02 AM
Plato
Quote:

Originally Posted by Thomas
I do notice that the second equation is double the first (except for $\displaystyle h$.)

You have that right.
So what is $\displaystyle h$?
• Sep 17th 2007, 06:58 AM
Jhevon
Quote:

Originally Posted by Thomas
That's what I'm asking... I'm not sure.

I do notice that the second equation is double the first (except for $\displaystyle h$.)

as Plato hinted, you're on the right track. remember, we have infinitely many solutions if we have some parameter replacing one of our variables. this means there can be no leading 1 in the column that corresponds to that variable. what is the link between that property and one line being a multiple of another?
• Sep 17th 2007, 05:28 PM
Thomas
Okay, so make the bottom row 0's?

Do I want the top row to be zero's as well? Or do I reduce the top row as normal? I'm thinking reduce as normal...
• Sep 17th 2007, 05:35 PM
Jhevon
Quote:

Originally Posted by Thomas
Okay, so make the bottom row 0's?

yes. what does h have to be to do that?

Quote:

Do I want the top row to be zero's as well? Or do I reduce the top row as normal? I'm thinking reduce as normal...
how exactly would you make the top row zeros? and if somehow you managed to do that, how would that help?
• Sep 17th 2007, 05:40 PM
Thomas
Quote:

Originally Posted by Jhevon
yes. what does h have to be to do that?

how exactly would you make the top row zeros? and if somehow you managed to do that, how would that help?

$\displaystyle h$ would need to be -14, correct?

You COULD make the top row zero's by subtracting half the bottom row. Not sure why I would - just a question.
• Sep 17th 2007, 05:42 PM
Jhevon
Quote:

Originally Posted by Thomas
$\displaystyle h$ would need to be -14, correct?

yes
• Sep 17th 2007, 05:52 PM
Thomas
Thank you for the help. :)