Question

If both $A-\frac{1}{2}I$ and $A+\frac{1}{2}I$ are othogonal 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

$(A^{\prime }-\frac{1}{2}I)(A-\frac{1}{2}I)=I$ and $(A^{\prime }+\frac{1}{2}I)(A+\frac{1}{2}I)=I$
$\Rightarrow A+A^{\prime }=0$
$\Rightarrow A^{\prime }=-A$
$\Rightarrow A^2=-\frac{3}{4}I$
or ${\left(\frac{-3}{4}\right)}^n=(det(A){)}^2$
Hence, $n$ is even.