Question

"---Select---"

A

B

I

A+B

A-B

DETAILS LARLINALG8 2.R.042. = 4-6 Let A, B, and A + B be nonsingular matrices. Prove that A-1 + 8-1 is nonsingular by showing that (A-1 +8-1-1 = A(A + B)-18. (A-1 + B-1)(A(A + B)-4B) = A-4(4( ---Select-- ~ 2)*8) +B={(A(A + B)-'B) |--Select... )(A + B)+48 + (---Select... ~) 'A(A + B)=48 ---Select-- ~ ) (A + B)-18+ (-Select-- ~ - -) ACA + B)-18 2) 'A)(A + B)=18 7)")(A + B)-48 ) (A + B)-8 :8-(( --Select-- ~ )(A + B)-1) -Select- + -Select- --Select- B+ -Select- -8°48 B+ Select- --Select- =B-( --Select- =)

Answer 1

) Let AB and A+B be hon singular So AS, 3-4, (A + B) " exists. that (A' +BU) A (A+B)«'B . we prove Now, (At B ' CALA+B)"'B) = AVA(A+BY+B) + B''(A(A+B)-TB) AYA (A+B)-B + BOIA (A+BJ-'B = I (A +B)413 + B A (A + B)-B I + B-1A) (A+B)YB (B-1B + B- B-IA) (A + B)-4B =B-(B + A) (A+B)-7B = B- (A+B) (A+B) B Brl [B BB = [ Then (A+354) A (A+B) B.