Let G={ [[a,b][c,d]] | a,b,c,d are integers} and H = { [[a,b][c,d]] | a+b+c+d = 0}. Show that if x is in H...

Let G={ [[a,b][c,d]] | a,b,c,d are integers} and H = { [[a,b][c,d]] | a+b+c+d = 0}. Show that if x is in H then x^(-1) is in H.

Let G={ [[a,b][c,d]] | a,b,c,d are integers} and H = { [[a,b][c,d]] | a+b+c+d = 0}. Show that if x is in H...
If you can explain what you mean by your use of square brackets here (ie. what does [[a,b][c,d]] represent?), then I'm prepared to take a crack at this.





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Forget my previous answer, we're dealing with matrices with integer entries as a group under matrix addition. Ok, so for a matrix x = [[a, b][c, d]] in H, its additive inverse is given by





x^-1 = [[-a, -b][-c, -d]].





Clearly, x^-1 also has integer entries. Adding the entries of x^-1 yields





-a + (-b) + (-c) + (- d) = - (a + b + c + d) = - 0 = 0.





Hence, x^-1 is also in H.