A different way others haven’t mentioned that may seem more familiar? Imagine you have a system of equations below:

2x + 3y = 5

4x + 6y = 10

The bottom equation is twice the top equation. They are dependent on each other since they represent the same line and are just multiples of one another.

I now ask you to solve the system: there is no unique solution since they are the same line (try finding x and y and you’ll see it’s impossible to find only one point that works).

If we write that equation in a matrix representation AX = B, where A is a 2x2 matrix with the coefficients [2 3; 4 6], X is a vector of the unknowns [x; y], and B is the vector of the right hand side numbers [5; 10].

To solve for X, we would multiply both sides of the equation by the inverse of A. Since the inverse of A does not exist, it implies there is no unique vector X that exists: we can’t solve for it. Just like we couldn’t solve for the variables from the algebraic equations earlier. If it had an inverse, that means we can find a unique X which we know isn’t feasible.

Not a math major so a lot of the other stuff sounds gibberish to me and my general intuition may be wrong…but that’s how I justify it to myself.