Find unknown element in a matrix

[RedXIII]

Homework Statement

Find all values of a for which the columns of the matrix are linearly dependent.

<br /> \left( {\begin{array}{*{20}c}<br /> 2 &amp; 0 &amp; 8 \\<br /> { - 4} &amp; 7 &amp; a \\<br /> 1 &amp; { - 3} &amp; 4 \\<br /> \end{array}} \right)<br />

Homework Equations

The Attempt at a Solution

So I took of rref of the matrix and found

<br /> \left( {\begin{array}{*{20}c}<br /> 1 &amp; 0 &amp; 4 \\<br /> { - 0} &amp; 1 &amp; 0 \\<br /> 0 &amp; { 0} &amp; a+16 \\<br /> \end{array}} \right)<br />

So then I have
a+16=0
a = -16


So that gives me 1 value which makes the columns of the matrix linearly dependent, but how do I find the other ones, or even know if more exist?

Thanks

 

[Dick]

Homework Statement

Find all values of a for which the columns of the matrix are linearly dependent.

<br /> \left( {\begin{array}{*{20}c}<br /> 2 &amp; 0 &amp; 8 \\<br /> { - 4} &amp; 7 &amp; a \\<br /> 1 &amp; { - 3} &amp; 4 \\<br /> \end{array}} \right)<br />

Homework Equations

The Attempt at a Solution

So I took of rref of the matrix and found

<br /> \left( {\begin{array}{*{20}c}<br /> 1 &amp; 0 &amp; 4 \\<br /> { - 0} &amp; 1 &amp; 0 \\<br /> 0 &amp; { 0} &amp; a+16 \\<br /> \end{array}} \right)<br />

So then I have
a+16=0
a = -16


So that gives me 1 value which makes the columns of the matrix linearly dependent, but how do I find the other ones, or even know if more exist?

Thanks

You've got it. There is just the one value, a=(-16). Otherwise the columns of the reduced matrix are linearly independent.