1. Lin. Dep. Matrix

Find all values of $\displaystyle a$ such that the set \displaystyle \{\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 \displaystyle \left[ \begin {array}{cc} a&a+2\\\noalign{\medskip}1&a\end {array}\right]

Row 1 - Row 2 yields:

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

Now not sure where to proceed.

2. 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 ?!

3. Hello,

Hmm I'll try for this one. Seems easy

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.

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

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

so now find a

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