If `A=1/3|[1, 2, 2], [2, 1,-2],[a,2,b]|` is an orthogonal matrix, then `a=-2` b. `a=2,b=1` c. `b=-1` d. `b=1`

Options

A. `a=2,b=1`
B. `a=-2,b=-1`
C. `a=2,b=-1`
D. `a=-2,b=1`

Answer

B

Explanation

Since, A is an orthogonal matrix.
`:. A A^T=I`
`rArr1/3{:[(1,2,2),(2,1,-2),(a,2,b)]. 1/3[(1,2,a),(2,1,2),(2,-2,b)]=[(1,0,0),(0,1,0),(0,0,1)]:}`
`rArr1/9{:[(1,2,2),(2,1,-2),(a,2,b)][(1,2,a),(2,1,2),(2,-2,b)]=[(1,0,0),(0,1,0),(0,0,1)]:}`
`rArr {:[(9,0,a+4+2b),(0,9,2a+2-2b),(a+4+2b,2a+2-2b,a^2+4+b^2)]=[(9,0,0),(0,9,0),(0,0,9)]:}`
`rArr a+4+2b=0,2a+2-2 b=0and a^2+4+b^2=9`
`rArra+2b+4=0,a-b+1=0anda^2+b^2=5`
`rArr a=-2,b=-1`