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Question

If the matrix 02βΥαβΥαβΥis orthogonal, then

A
α=±12
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B
β=±16
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C
γ=±13
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D
all of these
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Solution

The correct option is D all of these
02βγαβγαβγ
for orthogonal matrix we have
A.AT=I
02βγαβγαβγ0αα2βββγγγ=100010001
⎢ ⎢0+4β2+γ20+2β2γ202β2+γ20+2β2γ2α2+β2+γ2α2β2γ202β2+γ2α2β2γ2α2+β2+γ2⎥ ⎥=100010001
4β2+γ2=1,2β2γ2=0,
4(γ22)+γ2=1 β2=r22
r2[3]=1
r=±13
2β2γ2=0,α2+β2+γ2+γ2=1,α2β2γ2=0
β2=γ22,α2+γ22+γ21=1
α2+3γ22=1
β2=16α2+32×13=1
β=±16 α=±12

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