If A is a 3 X 3 non-zero matrix such that $$A^{2} = 0$$ then determinant of $$[(1 + A)^{2}- 50A]$$ is equal to

Given A is a 3x3 non zero matrix and $$A^2$$ = 0. 

Lets the consider the simple nilpotent matrix A, A = $$\begin{bmatrix}0&1&0\\0&0&0\\0&0&0\end{bmatrix}$$ , which is non zero matrix and $$A^2=0$$ .

$$[(1 + A)^{2}- 50A]$$ = $$[(I + A)^{2}- 50A]$$
= $$[I]^2+[A]^2+2[I][A]-50[A]$$

As it is known that$$[A]^2=0$$ , $$[I]^2=I$$ & [I][A] = [A] . lets substitute that in above :
= [I] - 48[A]
= $$\begin{bmatrix}1&0&0\\0&1&0\\0&0&1\end{bmatrix}$$ $$-\ $$ $$\begin{bmatrix}0&48&0\\0&0&0\\0&0&0\end{bmatrix}$$
=  $$\begin{bmatrix}1&-48&0\\0&1&0\\0&0&1\end{bmatrix}$$

Now, the determinant of above resultant matrix = 1(1-0) + 48(0-0) + 0  = 1
                                                                        
Therefore, the answer is 1.