If A = \(\begin{bmatrix}\alpha &\beta \\-\beta &\alpha \end{bmatrix}\) and \(A+ { A }^{ T } \) = \(\begin{bmatrix}0&1\\1&0\end{bmatrix}\), then the value of \(\alpha and \beta \) is
Answer & explanation
Correct answer: option 3
A = \(\begin{bmatrix}\alpha &\beta \\-\beta &\alpha \end{bmatrix}\)
\( { A }^{ T } \) = \(\begin{bmatrix}\alpha &-\beta \\\beta &\alpha \end{bmatrix}\)
\(A+ { A }^{ T } \) = \(\begin{bmatrix}\alpha &\beta \\-\beta &\alpha \end{bmatrix}\) + \(\begin{bmatrix}\alpha &-\beta \\\beta &\alpha \end{bmatrix}\) = \(\begin{bmatrix}0&1\\1&0\end{bmatrix}\)
\(A+ { A }^{ T } \) = \(\begin{bmatrix}0 &2\beta\\-2\beta &0 \end{bmatrix}\) = \(\begin{bmatrix}0&1\\1&0\end{bmatrix}\)
Equating all the corresponding terms of both the matrices we get
2\(\beta \) = 1 and so \(\beta \) = \(\frac{1}{2}\)
and also
\(\alpha - \alpha = 0\)
and so we cannot determine the values of the first variable.