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Question

If x,y,z are three real numbers and A=1cos(xy)cos(xz)cos(yx)1cos(yz)cos(zx)cos(zy)1
then
A is

A
symmetric matrix
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B
non-singular matrix
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C
not invertibe matrix
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D
orthogonal matrix
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Solution

The correct options are
A symmetric matrix
C not invertibe matrix
x,y,zR
A=1cos(xy)cos(xz)cos(yx)1cos(yz)cos(zx)cos(zy)1

A=ATcos(θ)=cosθ
A is symmetric

A=1cosxcosy+sinxsinycosxcosz+sinxsinz)cosycosx+sinysinx1cosycosz+sinysinzcoszcosx+sinzsinxcoszcosy+sinzsiny1

A=⎢ ⎢cos2x+sin2xcosxcosy+sinxsinycosxcosz+sinxsinz)cosycosx+sinysinxcos2y+sin2ycosycosz+sinysinzcoszcosx+sinzsinxcoszcosy+sinzsinycos2z+sin2z⎥ ⎥

A=cosxsinx0cosysiny0coszsinz0cosxcosycoszsinxsinysinz000

A=A1×A2|A1|=0 , |A2|=0

A is singular as |A|=|A1|.|A2|=0
A is not invertible
AAT=A.AA2I
A is not orthogonal

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