SOLUTION: Suppose that {{{A}}}, {{{B}}} and {{{A+B}}} are orthogonal matrices. Prove that {{{ (A+B)^-1 =A^-1 + B^-1}}} .

Algebra ->  Matrices-and-determiminant -> SOLUTION: Suppose that {{{A}}}, {{{B}}} and {{{A+B}}} are orthogonal matrices. Prove that {{{ (A+B)^-1 =A^-1 + B^-1}}} .      Log On


   

Question 1101998: Suppose that A, B and A%2BB are orthogonal matrices. Prove that
+%28A%2BB%29%5E-1+=A%5E-1+%2B+B%5E-1 .
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Given Fact #1: Matrix A is orthogonal. One property of orthogonal matrices is A%5E%28-1%29+=+A%5ET where A%5ET is the transpose of matrix A
Given Fact #2: Matrix B is orthogonal. Similar to Fact #1, we know that, B%5E%28-1%29+=+B%5ET
Given Fact #3: Matrix A%2BB is orthogonal. So we can say %28A%2BB%29%5E%28-1%29+=+%28A%2BB%29%5ET similar to the previous two facts above.


So,
%28A%2BB%29%5E%28-1%29+=+%28A%2BB%29%5ET (Fact #3)
%28A%2BB%29%5E%28-1%29+=+A%5ET%2BB%5ET Use the idea that %28A%2BB%29%5ET+=+A%5ET%2BB%5ET (transpose of a sum is equal to the sum of transposes)
%28A%2BB%29%5E%28-1%29+=+A%5E%28-1%29%2BB%5ET Substitution (Fact #1)
%28A%2BB%29%5E%28-1%29+=+A%5E%28-1%29%2BB%5E%28-1%29 Substitution (Fact #2)
And that's all there is to it