Question

Suppose $A$ is a $3 \times 3$ skew-symmetric matrix. Let $B=(I+A)^{-1}(I-A)$. Then

• A. $B$ is orthogonal
• B. $B$ is skew symmetric
• C. $B ^2= O$
• D. $B$ is a diagonal matrix

Answer 1

Answer: (A)

We have
$B B^{\prime} =(I+A)^{-1}(I-A)\left[(I+A)^{-1}(I-A)\right]^{\prime} $
$ =(I+A)^{-1}(I-A)(I-A)^{\prime}\left((I+A)^{\prime}\right]^{-1} $
$ =(I+A)^{-1}(I-A)(I+A)(I-A)^{-1} $
$ \left.=(I+A)^{-1}(I+A)(I-A)(I-A)^{-1} A^{\prime}=-A\right] $
$ =(I)(I)=I [I+A \text { and } I-A \text { commute }]$
$[I+A$ and $I-A$ commute $]$ is equal to:
Thus, $B$ is orthogonal.