Question

If both $A-\frac{1}{2}I$ and $A+\frac{1}{2}I$ are orthogonal matrices, then

• A. $A$ is orthogonal
• B. $A^2=\frac{3}{4}I$
• C. $A$ is skew-symmetric
• D. none of these

Answer 1

The correct option is C $A$ is skew-symmetric

For orthogonality of matrix $B:B{B}^T=I$
$\Rightarrow (A-\frac{1}{2}I){(A-\frac{1}{2}I)}^T=I$

$\Rightarrow A{A}^T-\frac{1}{2}{A}^T-\frac{1}{2}A+\frac{1}{4}I=I\cdots \left(i\right)$
and
$(A+\frac{1}{2}I){(A+\frac{1}{2}I)}^T=I$

$\Rightarrow A{A}^T+\frac{1}{2}{A}^T+\frac{1}{2}A+\frac{1}{4}I=I\cdots \left(ii\right)$

Subtracting $\left(i\right)$ from $\left(ii\right)$,
We get,
$A^T=-A$
So, $A$ is skew-symmetric.
Now, adding both equations :
We get, $A{A}^T=\frac{3I}{2}$
So, $A$ is not an orthogonal matrix.
and $A(-A)=\frac{3I}{2}$
$\Rightarrow A^2=-\frac{3I}{2}$