# Lin. Dep. Matrix

• Sep 29th 2008, 04:31 PM
LinAlg
Lin. Dep. Matrix
Find all values of $a$ such that the set \{\left[ \begin {array}{c} a\\\noalign{\medskip}1\end {array} \right], \left[ \begin {array}{c} a+2\\\noalign{\medskip}a\end {array}\right]\} is linearly dependent.

Work for this problem:

We have \left[ \begin {array}{cc} a&a+2\\\noalign{\medskip}1&a\end {array}\right]

Row 1 - Row 2 yields:

\left[ \begin {array}{cc} a&a+2\\\noalign{\medskip}a-1&2\end {array}\right]

Now not sure where to proceed.
• Sep 30th 2008, 10:55 AM
LinAlg
I believe a = -1 works just by playing with it, but I don't know how to tell if that's the only solution or if there are others. . . anyone ?!
• Sep 30th 2008, 11:09 AM
Moo
Hello,

Hmm I'll try for this one. Seems easy (Tongueout)

Two vectors are linearly dependent if there are m and n different from 0 such that mu+nv=0. ---> mu=-nv.
So you have to set a such that the coordinates in the matrices have a common ratio.

$\frac a1=\frac{a+2}{a}$

$\implies a^2=a+2 \implies a^2-a-2=0$

so now find a :)

Note : this is similar to finding a such that the matrix 2x2 you've got is non invertible (determinant = 0)