Question

If A=(1)/(3) beginbmatrix 1 & 2 & 2
2 & 1 & -2
a & 2 & b endbmatrix is an orthogonal matrix, then:

• A. \(a=-2,b=-1\)
• B. \(a=2,b=1\)
• C. \(a=2,b=-1\)
• D. a=-2,b=1

Hint:

Orthogonal matrices have mutually orthogonal rows of unit length.

Answer 1

The Correct Option is C

Step 1: For orthogonal matrix, rows are mutually perpendicular.
Step 2: Dot product of first and third row: 1· a+2·2+2· b=0 ⟹ a+4+2b=0 (1)
Step 3: Dot product of second and third row: 2a+1·2+(-2)b=0 ⟹ 2a+2-2b=0 (2)
Step 4: Solving (1) and (2): a=2, b=-1