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