Is the System {S,+,.} a Field?

[sam_0017]

true or false ..
The system {S,+,.} with S = { matrix (a,b,a-b,a)|a,b ∊ R)
is not a field under matrix addition (+) and matrix multiplication (.)





i find that the statement is false .
since : 1. {S,+} is Abelian group.
2. {S,.} : is Abelian group.

is my finding is true ?

 

[dodo]

Hi, Sam,
if by matrix(a,b,a-b,a) you mean$$\left( \begin{array}{cc} a & b \\ a-b & a \end{array} \right)$$and multiplication is the usual matrix multiplication, then try to multiply two such matrices, to see that the result may not have the same form. For example,$$\left( \begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array} \right)^2 = \left( \begin{array}{cc} 5 & 4 \\ 4 & 5 \end{array} \right)$$but 5-4 is not 4.