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Determinants

Question
CBSEENMA12036030
Wired Faculty App

A = open square brackets table row 1 2 2 row 2 1 cell negative 2 end cell row straight a 2 straight b end table close square brackets is a matrix satisfying the equation AAT = 9I, Where I is 3 x 3 identity matrix, then the ordered pair (a,b) is equal to

  • (2,-1)

  • (-2,1)

  • (2,1)

  • (-2,-1)

Solution

D.

(-2,-1)

Given, 
straight A space equals space open square brackets table row 1 2 2 row 2 1 cell negative 2 end cell row straight a 2 straight b end table close square brackets straight A to the power of straight T space equals space open square brackets table row 1 2 straight a row 2 1 cell negative 2 end cell row 2 cell negative 2 end cell straight b end table close square brackets AA to the power of straight T space equals space open square brackets table row 1 2 2 row 2 1 cell negative 2 end cell row straight a 2 straight b end table close square brackets open square brackets table row 1 2 straight a row 2 1 cell negative 2 end cell row 2 cell negative 2 end cell straight b end table close square brackets equals space open square brackets table row 9 0 cell straight a plus 4 plus 2 straight b end cell row 0 9 cell space 2 straight a plus 2 minus 2 straight b end cell row cell straight a plus 4 plus 2 straight b end cell cell space 2 straight a plus 2 minus 2 straight b end cell cell space straight a squared plus 4 plus straight b squared end cell end table close square brackets.
It is given that,
open square brackets table row 9 0 cell space space space straight a plus 4 plus 2 straight b end cell row 0 9 cell space space space space 2 straight a plus 2 minus 2 straight b end cell row cell straight a plus 4 plus 2 straight b end cell cell space space 2 straight a plus 2 minus 2 straight b end cell cell space space space space straight a squared plus 4 plus straight b squared end cell end table close square brackets space equals space 9 open square brackets table row 1 0 0 row 0 1 0 row 0 0 1 end table close square brackets rightwards double arrow open square brackets table row 9 0 cell space space space straight a plus 4 plus 2 straight b end cell row 0 9 cell space space space space 2 straight a plus 2 minus 2 straight b end cell row cell straight a plus 4 plus 2 straight b end cell cell space space 2 straight a plus 2 minus 2 straight b end cell cell space space space space straight a squared plus 4 plus straight b squared end cell end table close square brackets space equals space 9 open square brackets table row 9 0 0 row 0 9 0 row 0 0 9 end table close square brackets
On comparing we get,
a+ 4 +2b = 0
a+ 2b = -4   ... (i)
2a + 2-2b = 0
a-b= -1    ... (ii)
a2 + 4 +b2 = 9  ... (iii)
On solving eqs. (i) and (ii) we get
a = - 2, b = - 1
Hence, (a,b) ≡ (-2,-1)

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