Question

Lets $A=\left[\begin{array}{cc}0& 5\\ -5& 0\end{array}\right]$ be a skew symmetric matrix and $I+A$ is non singular, then the matrix $B=(I-A)(I+A{)}^{-1}$ is

• A. an Orthogonal Matrix
• B. an Idempotent Matrix
• C. a Nilpotent Matrix
• D. Data Insufficient

Answer 1

Answer: (A)

an Orthogonal Matrix

$B=(I-A)(I+A{)}^{-1}$

$B^T=\left[\right(I+A{)}^{-1}{]}^T(I-A{)}^T$

$\Rightarrow B^T=\left[\right(I+A{)}^T{]}^{-1}(I-A{)}^T$

$\Rightarrow B^T=(I-A{)}^{-1}(I+A)$ since $A^T=-A$

$B^{-1}=(I+A)(I-A{)}^{-1}$

In this case commutativity holds, so,

$B^T=B^{-1}\Rightarrow B\text{is Orthogonal Matrix}$