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Rational expressions are fractions in which both the numerator and the denominator are polynomials. Much like fractions in basic arithmetic, rational expressions can be added, subtracted, multiplied, divided, and simplified. A key method of simplifying complex rational expressions is partial fraction decomposition.

This technique is particularly useful when we want to integrate a rational expression or solve complex algebraic equations. In partial fraction decomposition, we express the complex rational expression as a sum of simpler fractions, making it easier to work with. The example given in the exercise, \(\dfrac{8x^2}{(x-1)^2(x+1)}\), involves breaking down a higher-degree polynomial denominator into linear and quadratic factors.

When we decompose a rational expression into partial fractions, we are essentially creating a system of equations to find the unknown coefficients of these simpler fractions. A system of equations is a set of two or more equations that have a common solution. In our example, after clearing the denominators, the numerators on both sides of the equation must be equal for all x-values, which provides the basis for our system.

To solve this system, we typically look for values of the variable that simplify the equations, such as zeros of the denominator polynomial, because they help us to isolate each unknown coefficient. By substituting these strategic values of x, we can turn a complex algebraic problem into a more manageable one with a clear-cut solution.

Matrix methods provide a structured approach to solving systems of equations. Two primary techniques are useful here: the use of the inverse matrix and Gaussian elimination. To apply matrix methods, we first represent the system as a matrix equation, A*X = B, where A is the matrix of coefficients, X is the column matrix of variables, and B is the matrix of constants.

Using matrix algebra, if we can find the inverse of matrix A, denoted as A^-1, we solve for X by multiplying both sides of our matrix equation by A^-1, thus obtaining X = A^-1*B. This method is straightforward but may be computationally intensive, especially for large matrices.

Gaussian elimination is another matrix method, which is often quicker and more practical, especially by hand for smaller systems. It transforms the matrix into row echelon form, and eventually in reduced row echelon form, by performing elementary row operations. These operations include swapping rows, multiplying a row by a nonzero number, and adding a multiple of one row to another row.

In the exercise, Gaussian elimination helps us reduce the matrix to an identity matrix, where the right of the augmented matrix corresponds to the solutions for A, B, and C. We proceed through a series of steps that simplify and isolate each variable, gradually moving towards the solution. The beauty of Gaussian elimination lies in its simplicity and the way it systematically zeros out the coefficients to ultimately reveal the unknowns in the system.