Question

For k =$\frac{1}{\sqrt{50}}$, the value of a, b, c such that PP¢ = I, where

P = $\begin{bmatrix} 2/3 & 3k & a \

• 1/3 & - 4k & b \ 2/3 & - 5k & c \end{bmatrix}$is-

• A. ±$\frac{16}{5\sqrt{2}}$, ±$\frac{13}{5\sqrt{2}}$, m$\frac{1}{3\sqrt{2}}$
• B. m$\frac{1}{3\sqrt{2}}$, ±$\frac{13}{5\sqrt{2}}$, ±$\frac{16}{5\sqrt{2}}$
• C. ±$\frac{13}{5\sqrt{2}}$, ±$\frac{16}{5\sqrt{2}}$, m$\frac{1}{3\sqrt{2}}$
• D. None of these

Answer 1

Answer: (C)

±$\frac{13}{5\sqrt{2}}$, ±$\frac{16}{5\sqrt{2}}$, m$\frac{1}{3\sqrt{2}}$

Answer 2

For PP¢ = 1,

$\begin{bmatrix} 2/3 & 3k & a \

• 1/3 & - 4k & b \ 2/3 & - 5k & c \end{bmatrix}$$\left[ \begin{array} { c c c } 2 / 3 & - 1 / 3 & 2 / 3 \ 3 \mathrm { k } & - 4 \mathrm { k } & - 5 \mathrm { k } \ \mathrm { a } & \mathrm { b } & \mathrm { c } \end{array} \right]$

= $\begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{bmatrix}$.Performing matrix multiplication, we have

$\frac{4}{9}$+ 9k² + a² = 1, $\frac{1}{9}$ + 16k² + b² = 1, $\frac{4}{9}$ + 25k² + c² = 0

Ž a² = $\frac{169}{450}$, b² = $\frac{256}{450}$, c² =$\frac{25}{450}$

Also $\frac{4}{9}$– 15k² + ac = 0, –$\frac{2}{9}$+ 20k² + bc = 0,

–$\frac{2}{9}$ – 12k² + ab = 0

Ž ab = $\frac{208}{450}$, bc = –$\frac{80}{450}$, ac = $\frac{- 65}{450}$.Hence a = ± $\frac{13}{5\sqrt{2}}$, b =± $\frac{16}{5\sqrt{2}}$, c m $\frac{1}{3\sqrt{2}}$.

Hence (3) is correct answer.