Matrices with infinitely many solutions

**Thomas** · September 16th 2007, 03:37 PM

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.

**Plato** · September 16th 2007, 03:55 PM

Gee, I think we need to see the exact question.

**Thomas** · September 16th 2007, 08:44 PM

Determine the value of $h$ such that the matrix is the augmented matrix of a linear system with infinitely many solutions.

$<br /> \begin{array}{c}\ \\ \\ \end{array}\;\begin{vmatrix}\;7 & -7 & 3\;\\\;14 & h & 6\end{vmatrix}<br />$

That's the exact question.

**Jhevon** · September 16th 2007, 09:13 PM

Originally Posted by Thomas

Determine the value of $h$ such that the matrix is the augmented matrix of a linear system with infinitely many solutions.

$<br /> \begin{array}{c}\ \\ \\ \end{array}\;\begin{vmatrix}\;7 & -7 & 3\;\\\;14 & h & 6\end{vmatrix}<br />$

That's the exact question.

ok. what do we need to have happen in order for there to be infinitely many solutions?

**Thomas** · September 17th 2007, 04:25 AM

That's what I'm asking... I'm not sure.

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

**Plato** · September 17th 2007, 05:02 AM

Originally Posted by Thomas

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

You have that right.
So what is $h$ ?

**Jhevon** · September 17th 2007, 06:58 AM

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 $h$ .)

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?

**Thomas** · September 17th 2007, 05:28 PM

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...

**Jhevon** · September 17th 2007, 05:35 PM

Originally Posted by Thomas

Okay, so make the bottom row 0's?

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

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?

**Thomas** · September 17th 2007, 05:40 PM

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?

$h$ 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.

**Jhevon** · September 17th 2007, 05:42 PM

Originally Posted by Thomas

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

yes

**Thomas** · September 17th 2007, 05:52 PM

Thank you for the help.

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