Take the tour ×  Suppose you have a linear transformation $T:
M_{2\times 2}\to M_{2\times 2}$ given by  $$ \begin{pmatrix} a &
b \\ c & d\end{pmatrix}\mapsto \begin{pmatrix} a+b & a \\ c &
c+d\end{pmatrix}$$  How can I tell quickly, if it is invertible?
share|improve this question add comment  3 Answers  up vote 1
down vote accepted  Hint: 1) You need to check the linear
transformation part.  2) If someone gives you an arbitrary
matrix, can you uniquely identify the matrix that it came from?
For example, let $$B=\begin{pmatrix} 12 & -3 \\ 4 &
\pi\end{pmatrix}.$$ What would be the only $A$ such that
$T(A)=B$? Note that "$a"$ must be $-3$, and $c$ must be $4$.
Thus $b$ must be $12-(-3)$ and $d$ must be $\pi -4$.
share|improve this answer Yes I suppose working backwards you
could. For example, if I gave the random matrix
$\begin{pmatrix}1 & 2 \\ 4 & 1\end{pmatrix}$ I could figure out
$a=2$ and $c=1$ etc.. and determine the original input matrix.
Is that right? If yes, is there a more general way to put it, fo