Determine whether $\displaystyle T$ is invertible, and compute $\displaystyle T^-1$ if it exists:

$\displaystyle T: M_{2x2}(R) \rightarrow R^4$ defined by

$\displaystyle T(A) = (tr(A), tr(A^T), tr(EA), tr(AE))$, where E = $\displaystyle \left({\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}} \right)$

so far I have:

$\displaystyle tr(A)= (1, 0, 0, 1) \\ tr(A^T) = (1, 0, 0, 1) \\ tr(EA) = (0, 1, 0, 1) \\ tr(AE) = (0, 1, 1, 0)$

= $\displaystyle \left({\begin{array}{cccc} 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \\ 1 & 1 & 1 & 0 \end{array}} \right)$

is this correct, or did I leave something out? I'm thinking that since the Rank is 3 and not 4, that this transformation is not invertible.