In numerical analysis:
Methods designed to find the roots (zeros) of nonlinear algebraic and transcendental equations \(f(x) = 0\) include:

• Bisection method (A) (bracketing method based on intermediate value theorem)
• Newton-Raphson method (B) (open iteration method using tangent slopes)
• Regula-Falsi / False Position method (D) (bracketing method using chords/secants)
• Secant method and Muller's method

In contrast, the Gaussian Elimination method (C) is a direct matrix method used to solve systems of simultaneous linear algebraic equations (\([A][X] = [B]\)), NOT for finding roots of a single nonlinear equation.

Using the chain rule of differential calculus:
Let

Since

Evaluating the definite integral using the power rule of integration:

For

Detailed Proof of Matrix Inversion Reversal Law:

• Reversal Law for Inverse: For two non-singular invertible square matrices $A$ and $B$, the product inverse satisfies $(AB)^{-1} = B^{-1} A^{-1}$.
• Verification: By definition of matrix inverse, multiplying $(AB)$ by $(B^{-1} A^{-1})$:
  $$(AB)(B^{-1} A^{-1}) = A(B B^{-1})A^{-1} = A(I)A^{-1} = A A^{-1} = I$$
  Similarly:
  $$(B^{-1} A^{-1})(AB) = B^{-1}(A^{-1} A)B = B^{-1}(I)B = B^{-1} B = I$$