Correct Answer - Option 1 :
\(- \frac{4}{5}\)
Concept:
Orthogonal matrix:
For an orthogonal matrix, transpose is equal to its inverse.
[M]T = [M-1]
For an orthogonal matrix:
MMT = I
Calculation:
Given:
[M] \(= \left[ {\begin{array}{*{20}{c}} {\frac{3}{5}}&{\frac{4}{5}}\\ x&{\frac{3}{5}} \end{array}} \right]\;\),
Given M is the orthogonal matrix
∴ [M][M]T = I
\(\left[ {\begin{array}{*{20}{c}} {\frac{3}{5}}&{\frac{4}{5}}\\ x&{\frac{3}{5}} \end{array}} \right]\left[ {\begin{array}{*{20}{c}} {\frac{3}{5}}&x\\ {\frac{4}{5}}&{\frac{3}{5}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right]\;\)
\(= \left[ {\begin{array}{*{20}{c}} {1}&{\frac{3x}{5}} +\frac{12}{25}\\ \frac{3x}{5}+\frac{12}{25}&x^2+{\frac{9}{25}} \end{array}} \right] = \left[ {\begin{array}{*{20}{c}} 1&0\\ 0&1 \end{array}} \right]\;\)
so, \(\frac{{12}}{{25}} + \frac{3}{5}x = 0 \Rightarrow x = -\frac{{4}}{5}\;\)
The value of x is \(\frac{{ - 4}}{5}\)