Question

If both $(A-\frac{I}{2})$ and $(A+\frac{I}{2})$ are orthogonal matrices, then

• A. A is orthogonal
• B. A is skew symmetric matrix of even order
• C. $A^2=\frac{3}{4}I$
• D. A is skew symmetric of odd order

Answer 1

The correct option is B A is skew symmetric matrix of even order

$(A-\frac{I}{2})(A^T-\frac{I}{2})=I$

$A{A}^T-\frac{A^T}{2}-\frac{A}{2}=\frac{3I}{4}$ ........ (1)

Similarly $A{A}^T+\frac{A^T}{2}+\frac{A}{2}=\frac{3I}{4}$ ........ (2)

equation (2) - equation (1) $\Rightarrow A+A^T=0\Rightarrow$ Skew symmetric matrix

(1) + (2) $\Rightarrow A{A}^T=\frac{3I}{4}$

So $|A|\ne 0$