Find all real matrices #A# , such that #A²= I(2)# (#A# is a matrix of second order)?
2 Answers
Solutions:
#((1,0),(0,1))# ,#((-1,0),(0,-1))# ,#((1,0),(c,-1))# ,#((-1,0),(c,1)),((a, b),((1-a^2)/b,-a))#
Explanation:
Suppose
Then:
#A^2 = ((a, b),(c, d))((a, b),(c, d)) = ((a^2+bc, b(a+d)),(c(a+d), d^2+bc))#
So if we want
#{ (a^2+bc = 1), (b(a+d) = 0), (c(a+d) = 0), (d^2+bc = 1) :}#
From the second equation, we have
Case
#a=+-1# ,#d=+-1#
If additionally
So the case
#((1,0),(0,1))# ,#((-1,0),(0,-1))# ,#((1,0),(c,-1))# ,#((-1,0),(c,1))#
Case
#bc = 1 - a^2#
This results in solutions:
#((a, b),((1-a^2)/b,-a))#
All of these solutions work.
See below
Explanation:
Given
we need all
where
Solving the system of resulting equations
we have
For