If both ( A - I 2 ) and A + I 2 are orthogonal matrices, then - Vidyadip

MCQ

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

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

✓

Answer

Correct option: B.

$A$ is skew symmetric matrix of even order

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

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

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

$(2)-(1) \Rightarrow A+A^{T}=0$

Skew symmetric matrix

But $(1)+(2)$

$\Rightarrow \mathrm{AA}^{\mathrm{T}}=\frac{3 \mathrm{I}}{4}$

But $|A| \neq 0$

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