Question

If both $A-\frac{1}{2}I$ and $A+\frac{1}{2}I$ 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. none of these

Answer 1

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

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

$\Rightarrow A{A}^T-\frac{1}{2}{A}^T-\frac{1}{2}A+\frac{1}{4}I=I$

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

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

Subtracting the two equations, we get,

$A^T=-A$

So, $A$ is skew-symmetric