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linear algebra with inverse of matrix.
Posted:
Mar 16, 2013 11:36 AM


Hello teacher~
nonzero 2 by 2 matrix A, B
A^2 + B^2 = 0 and (A+B)^2 = 0
(E : Identity matrix)
(1) AB = BA
(2) (A^3)(B^3) = (B^3)(A^3)
(3) A+B+E has an inverse matrix.
 (1) (A+B)^2 = (A+B)(A+B) = A^2 + AB + BA + B^2 = 0
Sinc A^2 + B^2 = 0, AB + BA = 0 so, AB = BA
(2) Since AB = BA, (A^3)(B^3) = AAABBB = AA(BA)BB = AA(B)(BA)B =AA(B)(B)(BA) = AA(B^3.A) It means that ABBB = BBBA so, AAABBB = AABBBA = ABBBAA = BBBAAA = (B^3)(A^3)
(3) Answer : inverse matrix ==> EAB How do you find it ?
Of course, it's really true. (A+B+E)(EAB) = AAAAB+BBABB+EAB = E



