Question

If `A=[1 2 2 2 1-2a2b]` is a matrix satisfying the equation `AA^T=""9I` , where `I` is `3xx3` identity matrix, then the ordered pair (a, b) is equal to : (1) `(2,-1)` (2) `(-2,""1)` (3) (2, 1) (4) `(-2,-1)`
A. (2, 1)
B. (-2, -1)
C. `(2, -1)`
D. `(-2, 1)`

Answer 1

Correct Answer - B
`therefore A A^(T) = 9 I`
`[[1,2,2],[2,1,-2],[a,2,b]][[1,2,a],[2,1,2],[2,-2,b]]= 9 [[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]]`
On comparing, we get
`a + 2b + 4 = 0 " " (i)`
` 2a- 2b+2=0" "(ii)`
From Eqs. (i) and (ii), we get
`a = -2,`
`b= -1`
`therefore` Prdered pair is `(-2, -1)`.