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Question

If A=13⎡⎢⎣12221−2a2b⎤⎥⎦ is an orthogonal matrix, then

A
a=2
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B
a=2,b=1
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C
b=1
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D
b=1
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Solution

The correct options are
A a=2
D b=1
For an orthogonal matrix A.AT=AT.A=I

Therefore, 13122212a2b×1312a21222b=100010001

1990a+4+2b092a+22ba+4+2b2a+22ba2+4+b2=100010001

Therefore, we get
a+4+2b9=0
a+2b=4 ...(i)

2a+22b9=0
2a+22b=0
a+1=b ...(ii)

and, a2+4+b29=1
a2+b2=5 ... (iii)

On solving equation (i),(ii) nd (iii),we get
a=2,b=1 or
a=1,b=2

Hence, the correct options are (A)and (C).

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