Algebraic Structures. How do i prove this 2x2 matrix isn't a group under matrix multiplication? (Prove only identity and/or inverse element..)
[tex]\displaystyle\\\text{Given}~M=\left\{\left[\begin{array}{ccc}a&b\\c&d\end{array}\right]~|~a,b,c,d \in Q \right\}. \text{Prove}~(M,\times)~\text{isn't a group}[/tex]
The problem lies with non-invertible matrices, i.e. those with determinant 0.
In fact, in order for [tex]M[/tex] to be a group, you need that:
[tex]M[/tex] is closed under multiplication. And it is, because the multiplication of two 2x2 matrices with rational coefficient is still a 2x2 matrix with rational coefficients
There exists an element [tex]e \in M[/tex] such that [tex]Ae=eA[/tex] for all [tex]A\in M[/tex]. And again you have it, [tex]e[/tex] is the matrix with 1 on the diagonal and 0 elsewhere
Each element [tex]A \in M[/tex] must have an inverse [tex]A' \in M[/tex] such that [tex]AA'=A'A=e[/tex], and this is where this example fails. If [tex]A[/tex] isn't invertible (for example, pick the all-zeroes matrix), then you can't find a matrix [tex]A' \in M[/tex] such that [tex]AA'=A'A=e[/tex].
