BITSAT Mathematics Question (2024) — Solution

Step-by-step solution

We know that, if A is an orthogonal matrix, then gathered \ AA^T=I \\ \\ [ array ccc 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b array ] 1 3 [ array ccc 1 & 2 & a \\ 2 & 1 & 2 \\ 2 & -2 & b array ]= [ array lll 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 array ] \\ [ array ccc 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b array ] [ array ccc 1 & 2 & a \\ 2 & 1 & 2 \\ 2 & -2 & b array ]= [ array lll 9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9 array ] \\ [ array ccc 9 & 0 & a+4+2 b \\ 0+4+2 b & 9 & 2 a+2-2 b \\ 2 a+2-2 b & a^2+4+b^2 array ] \\ = [ array ccc 9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9 array ] \\ a+4+2 b=0 and 2 a+2-2 b=0 \\ a+2 b=-4 (i)\\ a-b=-1 (ii) gathered By solving Eqs. (i) and (ii), we get a=-2, b=-1