See if system is linear I have this system $$y_{3} [n]= 2x[n-2]x[n-3]$$ and I have to see if the system is linear. I understand that I have to see if the system verifies the additivity and homogeneity properties. I'm having doubt's with the additivity so I came here to see if I'm doing it right or wrong.
I know that $T\left\{x_{1}[n]+x_{2}[n]\right \}=y_{3\_1}[n]+y_{3\_2}[n]$ so I've done this:
$y_{3}[n]=2(x[n-2]x[n-3]+x[n-2]x[n-3])=2x[n-2]x[n-3]+2x[n-2]x[n-3]=4x[n-2]x[n-3]$ and since this is different than $y_{3\_1}[n]+y_{3\_2}[n]$, the system is not linear.
For the homogeneity part I know that $T\left\{ax[n]\right\}=ay[n]$ so $y_{3}[n]=a(x[n-2]x[n-3])\Rightarrow y_{3}[n]=2x[n-2]x[n-3]$
I don't know if this is the correct way because just started to learn.

The system is not linear and in fact it fails both additivity and homogeneity. To show that it is nonlinear, it is enough to show that it fails at least one of the above two properties. Consider as input a unit step function $x[n \ge 0]=1, x[n<0]=0$. Then $y[n \ge 3] = 2$. Now consider as input the unit step function multiplied by $\alpha \neq 1,0$. Then $y[n \ge 3] = 2 \alpha^2 \neq 2 \alpha$. Hence homogeneity does not hold.
Your argument regarding additivity is not quite sound. The way to proceed is to consider an input $u[n]=x_1[n]+x_2[n]$ and then show that $2x_1[n-2]x_1[n-3]+2x_2[n-2]x_2[n-3] \neq 2 (x_1[n-2]+x_2[n-2])(x_1[n-3]+x_2[n-3])$.
