if              is a matrix satisfying the equation AAT=9I, where I is 3x3 identity matrix, then the ordered pair

(a, b) is equal to :

  • Option 1)

    (2,-1)

  • Option 2)

    (-2,1)

  • Option 3)

    (2,1)

  • Option 4)

    (-2,-1)

 

Answers (1)

Using

Multiplication of matrices -

-

 

 \begin{bmatrix} 1 & 2 & 2\\ 2 & 1 & -2\\ a & 2 & b \end{bmatrix}\begin{bmatrix} 1 & 2 & a\\ 2 & 1 & 2\\ 2 & -2 & b \end{bmatrix}=\begin{bmatrix} 9 & 0 & 0\\ 0 & 9 & 0\\ 0 & 0 & 9 \end{bmatrix}               

by given condition

\Rightarrow\begin{bmatrix} 9 & 0 & a+4+2b\\ 0 & 9 & 2a+2-2b\\ 0 & 0 & a^{2}+4+b^{2} \end{bmatrix}-\begin{bmatrix} 9 & 0 & 0\\ 0 & 9 & 0\\ 0 & 0 & 9 \end{bmatrix}

\therefore a=-2

b=-1


Option 1)

(2,-1)

This option is incorrect.

Option 2)

(-2,1)

This option is incorrect.

Option 3)

(2,1)

This option is incorrect.

Option 4)

(-2,-1)

This option is correct.

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