For any 3x3 matrix M, let |M| denote the determinant of M. Let I be the 3 x 3 identity matrix. Let E and F be two 3x3 matrices such that (I - EF) is invertible. If G = (I - EF) -1 , then which of the following statements is(are) TRUE?
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(a, b, c)
G(I - EF) = (I - EF)G = I
G - GEF = G - EFG = I .....(1)
|FE| = |I - FE| |FGE| = |FGE - FE FGE|
= |FGE-F(G-I)E|=|FGE-FGE+FE|=|FE|
(I - FE)(I + FGE) = I + FGE - FE - FEFGH
= I + FGE - FE - F(G - I)E = I + FGE - FE - FGE + FE = I
From (I) it is true
(I - FE)(I - FGE) = I - FGE - FE + FEFGE
= I - FGE - FE + F(G - I)E = I - FGE - FE + FGE - FE
= I - 2FE
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