First slide

Transpose of matrix

Question

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

Moderate

A

B is orthogonal
B

B is skew symmetric
C

$B^{2} = O$
D

B is a diagonal matrix

Solution

We have
$\begin{array}{l} B B^{'} = (I + A)^{- 1} (I - A) {[(I + A)^{- 1} (I - A)]}^{'} \\ = (I + A)^{- 1} (I - A) (I - A)^{'} {((I + A)^{'}]}^{- 1} \\ = (I + A)^{- 1} (I - A) (I + A) (I - A)^{- 1} \end{array}$
$[∵ A^{'} = - A]$
$= (I + A)^{- 1} (I + A) (I - A) (I - A)^{- 1}$
[ $I + A$ and $I - A$ commute]
$= (I) (I) = I$
Thus, B is orthogonal.

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App