Question with linear transformation and matrices, to try to reach conclusions. Let $T:\mathbb{R}^{2\times 2} \to \mathbb{R}^{2\times 2}$.     $T(\begin{bmatrix}x&amp;y\\z&amp;w\end{bmatrix})$ = $\begin{bmatrix}-2x+y&amp;-y + z\\3w&amp;w\end{bmatrix} $. 
And there's a basis $B$ such that $[T]_B = \begin{bmatrix}a&amp;1&amp;0&amp;0\\0&amp;-2&amp;0&amp;0\\0&amp;0&amp;0&amp;b\\0&amp;0&amp;2&amp;-1\end{bmatrix}$. 
Then which of these statements is true? 
a) $a= 0$ and $b\ne 0$. 
b) $a = 1$ and $b=0$.
c) $a=-1$ and $b=1$. 
d) any $a$ and $b = 0$. 
e) $a=1$ and any $b$. 
My approach (The idea): I know that $ImT = Col[T]_B$, so I first found out that $dimImT=3$, so $Rank([T]_B)$ should be equal to $3$. so the answer isn't c,d,e. 
Now I got stuck in deciding if it's (a) or (b), since both of the answers lead to $Rank([T]_B) = 3$, and I couldn't figure out how to find the correct answer. 

I would really appreciate any hints or feedback, thanks in advance.
Consider the following  results:

for a linear operator $T$ rankT =rank of the associated matirx in some ordered basis say $B$.

Then from your calulations  you see the linear map has  rank  3  so its associated matrix in some ordered  basis has also rank 3
Now you just calculate the rank of the matrix that you given.
Try to find its row-reduceed echelon form or row reducedfrom to see its rank.
I am giving final matrix in the row reduceed echelon form in the below picture that i attached. Please find it.
From that you cansee  $b$ should  be non zero to have a  rank 3 .
Which is given in only option ( a). So it the answer.
You think why option ( c) is not the answer  ,as it will give the rank of the matrix 4.
For any other queries  you can put the values of a and b to find the rank of the matrix is 3 or 4
https://m.matrix.reshish.com/rankCalculation.php
Consider the following  results:

for a linear operator $T$ rankT =rank of the associated matirx in some ordered basis say $B$.

Then from your calulations  you see the linear map has  rank  3  so its associated matrix in some ordered  basis has also rank 3
Now you just calculate the rank of the matrix that you given.
Try to find its row-reduceed echelon form or row reducedfrom to see its rank.
I am giving final matrix in the row reduceed echelon form in the below picture that i attached. Please find it.
From that you cansee  $b$ should  be non zero to have a  rank 3 .
Which is given in only option ( a). So it the answer.
You think why option ( c) is not the answer  ,as it will give the rank of the matrix 4.
For any other queries  you can put the values of a and b to find the rank of the matrix is 3 or 4
https://m.matrix.reshish.com/rankCalculation.php