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 find the functions \( f(x) \) and \( g(x) \) from the given expression for \( e^{A} \), where \( A = \begin{pmatrix} x & x \\ x & x \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate \( A^2 \)
To find \( A^2 \), we multiply \( A \) by itself:
\[ A^2 = A \cdot A = \begin{pmatrix} x & x \\ x & x \end{pmatrix} \cdot \begin{pmatrix} x & x \\ x & x \end{pmatrix} \]
Calculating the elements:
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