Question

If 3A = $\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & –2 \\ x & 2 & y \end{bmatrix}$and A is orthogonal, then x + y =

• A. 3
• B. – 2
• C. –3
• D. None

Answer 1

Answer: (C)

–3

Answer 2

A = $\frac { 1 } { 3 }$ $\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & –2 \\ x & 2 & y \end{bmatrix}$Ž A = $\frac{1}{3}$B (Let)

and AA' = I

Ž$\left( \frac{1}{3}.B \right)$. $\left( \frac{1}{3}.B \right)^{'}$ = I

= B.B' = 9 I

Ž $\begin{bmatrix} 1 & 2 & 2 \\ 2 & 1 & –2 \\ x & 2 & y \end{bmatrix}$. $\begin{bmatrix} 1 & 2 & x \\ 2 & 1 & 2 \\ 2 & –2 & y \end{bmatrix}$ = $\begin{bmatrix} 9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9 \end{bmatrix}$

Ž x + 4 + 2y = 0, 2x + 2 – 2y = 0

Ž x = – 2, y = – 1

\ x + y = (–2) + (–1)

= –3