Algebraic Structures. How do i prove this 2x2 matrix isn't a group under matrix multiplication? (Prove only identity and/or inverse element..)

The problem lies with non-invertible matrices, i.e. those with determinant 0.

In fact, in order for to be a group, you need that:

is closed under multiplication. And it is, because the multiplication of two 2x2 matrices with rational coefficient is still a 2x2 matrix with rational coefficientsThere exists an element such that for all . And again you have it, is the matrix with 1 on the diagonal and 0 elsewhereEach element must have an inverse such that , and this is where this example fails. If isn't invertible (for example, pick the all-zeroes matrix), then you can't find a matrix such that .

So, this isn't a group

It’s b because vpt is the whole triangle and ptc isn’t

There is no question, there is just the answer.