If `e^(A)` is defined as `e^(A)=I+A+A^(2)/(2!)+A^(3)/(3!)+...=1/2 [(f(x),g(x)),(g(x),f(x))]`, where `A=[(x,x),(x,x)], 0 lt x lt 1` and I is identity matrix, then find the functions f(x) and g(x).

Explanation

To solve the problem, we need to find the functions \( f(x) \) and \( g(x) \) given that:

\[ e^{A} = I + A + \frac{A^2}{2!} + \frac{A^3}{3!} + \ldots = \frac{1}{2} \begin{pmatrix} f(x) & g(x) \\ g(x) & f(x) \end{pmatrix} \]

where \( A = \begin{pmatrix} x & x \\ x & x \end{pmatrix} \) and \( 0 < x < 1 \).

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