4a
1) If A and B are any two different square matrices of order n with A - B is non-singular
A3 = B3 and A(AB) = B(BA) , then
a) A2 + B2 = 0 b) A2 +B2 = I
c) A2 + B3 = I d) A3 + B3 = 0
-
UP 0 DOWN 0 0 4
4 Answers
62Since A-B is non singular,
(A-B)^3\neq 0, hence
\\(A-B)(A-B)^2=(A-B)(A^2-AB-BA+B^2)\neq 0 \\\Rightarrow A^3-A(AB)-A(BA)+A(BB)-B(AA)+B(AB)+B(BA)-B^3\neq 0 \\\Rightarrow -A(BA)+A(BB)-B(AA)+B(AB)\neq 0 \\\Rightarrow (AB)(-A+B)-BA(A-B)\neq 0 \\\Rightarrow (AB+BA)(A-B)\neq 0
\\(A-B)(A-B)^2=(A-B)(A^2-AB-BA+B^2)\neq 0 \\\Rightarrow A^3-A^2B-ABA+AB^2-BA^2+BAB+B^2A-B^3\neq 0 \\\Rightarrow -ABA+AB^2-BA^2+BAB\neq 0 \\\Rightarrow BA+AB\neq 0
Am i missing out something! :(