1. ## Linear dependence question

The question:
Determine, with reasons whether or not the matricies

$A_1 = $\left( \begin{array}{cc} 1 & 2 \\ 2 & 1\end{array} \right)$$

$A_2 = $\left( \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array} \right)$$

$A_3 = $\left( \begin{array}{cc} 2 & 1 \\ 2 & 1\end{array} \right)$$

are linearly dependent.

My attempt:
Usually with these types of questions I'd create a large matrix and reduce to row echelon form. So I'd start by stating:
$\lambda_1A_1 + \lambda_2A_2 + \lambda_3A_3$ = 0

And create a matrix from there. How do I do this when we have matrices instead of polynomials or vectors? Thanks!

2. Originally Posted by Glitch
The question:
Determine, with reasons whether or not the matricies

$A_1 = $\left( \begin{array}{cc} 1 & 2 \\ 2 & 1\end{array} \right)$$

$A_2 = $\left( \begin{array}{cc} 0 & 1 \\ 1 & 0\end{array} \right)$$

$A_3 = $\left( \begin{array}{cc} 2 & 1 \\ 2 & 1\end{array} \right)$$

are linearly dependent.

My attempt:
Usually with these types of questions I'd create a large matrix and reduce to row echelon form. So I'd start by stating:
$\lambda_1A_1 + \lambda_2A_2 + \lambda_3A_3$ = 0

And create a matrix from there. How do I do this when we have matrices instead of polynomials or vectors? Thanks!
If it helps (if it doesn't let me know) try considering that two by two matrices are just a fancy representation of four-tuples.

3. So you're saying to write the matrices as 4-tuples, then form a matrix etc?

I just tried the following:

Equating components:
$a_{11} = \lambda_1 + 0 + 2\lambda_3 = 0$
$a_{12} = 2\lambda_1 + \lambda_2 + \lambda_3 = 0$
$a_{21} = 2\lambda_1 + \lambda_2 + 2\lambda_3 = 0$
$a_{22} = \lambda_1 + 0 + \lambda_3 = 0$

Which gives us:

$$\left( \begin{array}{ccc} 1 & 0 & 2 \\ 2 & 1 & 1 \\ 2 & 1 & 2 \\ 1 & 0 & 1 \end{array} \right)$$

Is this correct? Thanks.

4. Is that approach correct? My final answer was that it was linearly independent. My book doesn't have solutions.

5. Yes, that is correct.