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3 and 4 .Determinants and Matrices
normal
If both $\left( {A - \frac{I}{2}} \right)$ 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 matrix of odd order
Solution
$\left(A-\frac{I}{2}\right)\left(A^{T}-\frac{I}{2}\right)=I$
$A{A^T} – \frac{{{A^T}}}{2} – \frac{A}{2} = \frac{{3I}}{4}$ …….$(1)$
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 $|A| \neq 0$
Standard 12
Mathematics
