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If A=[(1,2,2),(2,1,-2),(a,2,b)] is a mat...

If `A=[(1,2,2),(2,1,-2),(a,2,b)]` is a matrix satisfying `"AA"^T=9I_3` , then find the values of `a` and `b` .

Text Solution

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`A^T=[[1, 2, a], [2, 1, 2], [2, -2, b]]`
`AA^T=[[9, 0, a+4+2b], [0, 9, 2a+2-2b], [a+4+2b, 2a+2-2b, a^2+4+b^2]]`
`AA^T=9I`
`[[9, 0, a+4+2b], [0, 9, 2a+2-2b], [a+4+2b, 2a+2-2b, a^2+4+b^2]]=9[[1, 0, 0], [0, 1, 0], [0, 0,1]]`
`=[[9, 0, 0], [0, 9, 0], [0, 0, 9]]`
`a+4+2b=0=>a+2b=-4--(1)`
`2a+2-2b=0=>a-b=-1--(2)`
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