If \( A=\begin{bmatrix}\displaystyle\frac{3}{5} & \displaystyle\frac{4}{5}\\ x & \displaystyle\frac{3}{5}\end{bmatrix} \) & given that \( A^{T}=A^{-1} \), then the value of \( x \) is:
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{-4}{5} \)
The condition \( A^{T}=A^{-1} \) means \( AA^{T}=I \).
The off-diagonal entry of \( AA^{T} \) must be zero:
\( \displaystyle \frac{3}{5}\cdot x+\frac{4}{5}\cdot\frac{3}{5}=0 \)
\( \displaystyle x=-\frac{4}{5} \)
As a check, the second diagonal entry is \( x^{2}+\frac{9}{25}=\frac{16}{25}+\frac{9}{25}=1 \).