If \( A=\displaystyle\frac{1}{3}\begin{bmatrix}1 & 2 & 2\\ 2 & 1 & -2\\ a & 2 & b\end{bmatrix} \) satisfies \( AA^{T}=I_{3} \) (Identity matrix of order 3), then the value of \( a \) and \( b \) are :
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( a=-2,\qquad b=-1 \)
The condition \( AA^{T}=I \) means that the rows of \( 3A \) are mutually perpendicular, each with squared length 9.
Let \( R_1=(1,2,2) \), \( R_2=(2,1,-2) \) and \( R_3=(a,2,b) \). Then
\( R_1\cdot R_3=a+4+2b=0 \)
\( R_2\cdot R_3=2a+2-2b=0 \)
The second equation gives \( b=a+1 \). Substituting in the first,
\( 3a+6=0 \)
so \( a=-2 \) and \( b=-1 \).
As a check, \( a^{2}+4+b^{2}=4+4+1=9 \).