If \( A=\begin{bmatrix}1&2&2\\2&1&-2\\m&2&n\end{bmatrix} \) is a matrix, such that \( AA^{T}=9I_{3} \), then the values of \( m \) and \( n \) are (where \( I_{3} \) is an identity matrix of order 3)
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \( m=-2,\ n=-1 \)
The entry in row i, column j of \( AA^{T} \) is the dot product of row i and row j of A. Since \( AA^{T}=9I_{3} \), the rows of A are mutually perpendicular, and each has squared length 9.
The first two rows already satisfy this:
\( (1,2,2)\cdot(1,2,2)=9 \)
\( (2,1,-2)\cdot(2,1,-2)=9 \)
\( (1,2,2)\cdot(2,1,-2)=0 \)
The third row \( (m,2,n) \) must be perpendicular to both:
\( m+4+2n=0 \)
\( 2m+2-2n=0 \)
The second equation gives \( n=m+1 \). Putting this into the first,
\( 3m+6=0 \)
\( m=-2,\ n=-1 \)
Check: \( m^{2}+2^{2}+n^{2}=4+4+1=9 \), as needed.