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 $|A| \neq 0$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

If $y=y(x)$ is the solution curve of the differential equation $\left(x^2-4\right) d y-\left(y^2-3 y\right) d x=0,$ $x>2, y(4)=\frac{3}{2}$ and the slope of the curve is never zero, then the value of $y(10)$ equals :

Solution set of the inequality $2x + y\, >\, 5$ is $.......$

The equation of the curve passing through $(3 , 4) \,\&$ satisfying the differential equation,
$y {\left( {\frac{{dy}}{{dx}}} \right)^2} + (x - y) \frac{{dy}}{{dx}} - x = 0 $ can be

Given that ${d \over {dx}}f(x) = f\,'(x)$. The relationship $f\,'(a + b) = f\,'(a) + f\,'(b)$ is valid if $f(x)$ is equal to

If $f(x)\, = \,\mathop {\lim }\limits_{n \to \infty } {\mkern 1mu} \frac{{\left[ {{x^2}} \right] + [{{(2x)}^2}] + [{{(3x)}^2}] + \cdots + [{{(nx)}^2}]}}{{{n^3}}}{\mkern 1mu} $ then $f(x)$ is (Where $[·]$ is G.I.F.)

If $|z| = 2$, then the points representing the complex numbers $ - 1 + 5z$ will lie on a

The distance between two parallel lines $3x + 4y - 8 = 0$ and $3x + 4y - 3 = 0$, is given by

Consider $f(x) =$ $\left[ {\frac{{2\,\,\left( {\sin \,x\,\, - \,\,{{\sin }^3}\,x} \right)\,\, + \,\,\left| {\sin \,x\,\, - \,\,{{\sin }^3}\,x} \right|}}{{2\,\,\left( {\sin \,x\,\, - \,\,{{\sin }^3}\,x} \right)\,\, - \,\,\left| {\sin \,x\,\, - \,\,{{\sin }^3}\,x} \right|}}} \right]$ , $x \ne \,\frac{\pi}{2} \,for\, x \in (0, \pi ) f(\pi /2) = 3$

where [ ] denotes the greatest integer function then,

Let $A=\left(\begin{array}{ccc}{[x+1]} & {[x+2]} & {[x+3]} \\ {[x]} & {[x+3]} & {[x+3]} \\ {[x]} & {[x+2]} & {[x+4]}\end{array}\right),$ where $[t]$ denotes the greatest integer less than or equal to $\mathrm{t}$. If $\operatorname{det}(\mathrm{A})=192$, then the set of values of $\mathrm{x}$ is the interval

Let $S$ be the set of all $a \in N$ such that the area of the triangle formed by the tangent at the point $P ( b , c ), b , c \in N$, on the parabola $y ^2=2 ax$ and the lines $x=b, y=0$ is $16$ unit $^2$, then $\sum_{\text {aes }} a$ is equal to $..........$.