For the given orthogonal matrix Q, Q=[\table \frac{3}{7} , \frac{2}{7} , \frac{6}{7} ; -\frac{6}{7} , \frac{3}{7} , \frac{2}{7} ; \frac{2}{7} , \frac{6}{7} , -\frac{3}{7} ] The inverse is

Correct Answer is : Q=[37−67272737676727−37]

Q=[\table \frac{3}{7} , \frac{2}{7} , \frac{6}{7} ; -\frac{6}{7} , \frac{3}{7} , \frac{2}{7} ; \frac{2}{7} , \frac{6}{7} , -\frac{3}{7}]

Q=[\table -\frac{3}{7} , -\frac{2}{7} , -\frac{6}{7} ; \frac{6}{7} , -\frac{3}{7} , -\frac{2}{7} ; -\frac{2}{7} , -\frac{6}{7} , \frac{3}{7}]

Q=[\table \frac{3}{7} , -\frac{6}{7} , \frac{2}{7} ; \frac{2}{7} , \frac{3}{7} , \frac{6}{7} ; \frac{6}{7} , \frac{2}{7} , -\frac{3}{7}]

Q=[\table -\frac{3}{7} , \frac{6}{7} , -\frac{2}{7} ; -\frac{2}{7} , -\frac{3}{7} , -\frac{6}{7} ; -\frac{6}{7} , -\frac{2}{7} , \frac{3}{7}]

GATE Civil Engineering (CE) 2018 Shift 1 Solved Paper

Question : 32 of 65

Marks: +1, -0

For the given orthogonal matrix

Q

Q=[\table \;{3}/{7} , \;{2}/{7} , \;{6}/{7} ; -\;{6}/{7} , \;{3}/{7} , \;{2}/{7} ; \;{2}/{7} , \;{6}/{7} , -\;{3}/{7} ]

The inverse is

$Q=[\table \;{3}/{7} , \;{2}/{7} , \;{6}/{7} ; -\;{6}/{7} , \;{3}/{7} , \;{2}/{7} ; \;{2}/{7} , \;{6}/{7} , -\;{3}/{7}]$
$Q=[\table -\;{3}/{7} , -\;{2}/{7} , -\;{6}/{7} ; \;{6}/{7} , -\;{3}/{7} , -\;{2}/{7} ; -\;{2}/{7} , -\;{6}/{7} , \;{3}/{7}]$
$Q=[\table \;{3}/{7} , -\;{6}/{7} , \;{2}/{7} ; \;{2}/{7} , \;{3}/{7} , \;{6}/{7} ; \;{6}/{7} , \;{2}/{7} , -\;{3}/{7}]$
$Q=[\table -\;{3}/{7} , \;{6}/{7} , -\;{2}/{7} ; -\;{2}/{7} , -\;{3}/{7} , -\;{6}/{7} ; -\;{6}/{7} , -\;{2}/{7} , \;{3}/{7}]$

Solution:

|Q|=\;{3}/{7}(-\;{9}/{49}-\;{12}/{49})-\;{2}/{7}(\;{18}/{49}-\;{4}/{49})+\;{6}/{7}

(\;{-36}/{49}-\;{6}/{49})=-1

Adj.

Q=[\table -\;{21}/{49} , \;{42}/{49} , -\;{14}/{49} ; -\;{14}/{49} , -\;{21}/{49} , -\;{42}/{49} ; -\;{42}/{49} , -\;{14}/{49} , \;{21}/{49}]

∴ \;\; Q^{-1}=\;{A d j Q}/{|Q|}=Q=[\table \;{3}/{7} , -\;{6}/{7} , \;{2}/{7} ; \;{2}/{7} , \;{3}/{7} , \;{6}/{7} ; \;{6}/{7} , \;{2}/{7} , -\;{3}/{7}]

Go to Question:

Prev Question

Next Question