If \(3A = \begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & -2 \\ x & 2 & y \end{bmatrix}\) and \(AA^T = I\), then x + y is equal to
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → −3
Let B = 3A. Then
\(BB^T = 9AA^T = 9I\)
The (i, j) entry of \(BB^T\) is the dot product of row i and row j of B. So the rows of B are mutually perpendicular, each with squared length 9.
Row 1 · Row 3 = 0:
\(x + 4 + 2y = 0 \qquad (1)\)
Row 2 · Row 3 = 0:
\(2x + 2 - 2y = 0\)
\(x = y - 1 \qquad (2)\)
Putting (2) in (1):
\(y - 1 + 4 + 2y = 0\)
\(y = -1,\quad x = -2\)
Check: row 3 is (−2, 2, −1), and \(4 + 4 + 1 = 9\), as needed.
\(x + y = -2 + (-1) = -3\)