Is $y[n]=x[n] * x[n^2]$ invertible?

Verdict: non-invertible. This is a first way.

1. How do you know if an equation is invertible?
2. How do you show a function is invertible?
3. Which functions are not invertible?
4. How do you prove that a matrix is not invertible?

How do you know if an equation is invertible?

In general, a function is invertible only if each input has a unique output. That is, each output is paired with exactly one input. That way, when the mapping is reversed, it will still be a function!

How do you show a function is invertible?

A function f : A → B is said to be invertible if it has an inverse function. Notation: If f : A → B is invertible, we denote the (unique) inverse function by f-1 : B → A. -1 ◦ f = IA.

Which functions are not invertible?

A lot of them, for example y = x^2 is not invertible, because two different values of x correspond to the same value of y. Only monotonously increasing or decreasing functions are invertible, i.e., those whose derivative is either always positive or always negative.

How do you prove that a matrix is not invertible?

We say that a square matrix is invertible if and only if the determinant is not equal to zero. In other words, a 2 x 2 matrix is only invertible if the determinant of the matrix is not 0. If the determinant is 0, then the matrix is not invertible and has no inverse.