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if A&B are two square matrices such that B= -A -1 BA, then prove that ( A+B) 2 =A 2 +B 2

if A&B are two square matrices such that B= -A-1BA, then prove that ( A+B)2=A2+B2

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Grade:12th pass

2 Answers

Amit Sharma
33 Points
6 years ago
It  is given that   
                                B = -A-1BA
           now, pre-multiplying by A on both sides, we will get
                                 AB = -BA
                                AB+BA = 0.........................(1)
Now coming to the problem:
(A+B)2=(A+B)*(A+B)  =  A2+B2+AB+BA.....................(2)
from (1) and (2), we get
(A+B)2=(A+B)*(A+B)
 
Amit Sharma
33 Points
6 years ago
 
 
It  is given that   
                                B = -A-1BA
           now, pre-multiplying by A on both sides, we will get
                                 AB = -BA
                                AB+BA = 0.........................(1)
Now coming to the problem:
(A+B)2=(A+B)*(A+B)  =  A2+B2+AB+BA.....................(2)
from (1) and (2), we get
( A+B)2=A2+B2 as AB+BA =0 {from (1)}

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