Explanation

(d)
$A\begin{bmatrix}
1 \\
- 1
\end{bmatrix} = \begin{bmatrix}
- 1 \\
2
\end{bmatrix}$ (1)
$A^{2}\begin{bmatrix}
1 \\
- 1
\end{bmatrix} = \begin{bmatrix}
1 \\
0
\end{bmatrix}$ (2)
Let A be given by$A = \begin{bmatrix}
a & b \\
c & d
\end{bmatrix}$. The first equation gives
$a - b = - 1$ (3)
$c - d = 2$(4)
For second equation gives
$A^{2}\begin{bmatrix}
1 \\
- 1
\end{bmatrix} = A\left( A\begin{bmatrix}
1 \\
- 1
\end{bmatrix} \right) = A\left( \begin{bmatrix}
- 1 \\
2
\end{bmatrix} \right) = \begin{bmatrix}
1 \\
0
\end{bmatrix}$
This gives
$- a + 2b = 1$ (5)
$- c + 2d = 0$ (6)
Eqs. (3)+(5)$\ \Rightarrow b = 0$ and $a = - 1$
Eqs. (4)+(6)$\Rightarrow d = 2\$and$\ c = 4$
So the sum $a + b + c + d = 5.$