How can matrices represent the objective function and constraints of a linear programming problem?

In linear programming, matrices provide a compact, algebraic framework for representing optimization problems. This approach allows for efficient manipulation of variables and constraints.1

Objective Function

The objective function is represented using the product of two vectors:
* $x$: A column vector $(x_1,x_2,\dots ,x_n{)}^T$ containing the objective variables.
* $p$: A row vector $(p_1,p_2,\dots ,p_n)$ containing the coefficients for each objective variable.

The objective function is expressed as the product $px$, which aggregates the variables and their respective weights into a single algebraic term.1

Constraints

The system of inequality constraints is represented by the matrix equation $Ax\le b$:
* $A$: An $m\times n$ matrix where the $m$ rows represent the individual constraints and the $n$ columns correspond to the objective variables.
* $b$: A column vector $(b_1,b_2,\dots ,b_m{)}^T$ containing the inequality bounds for each constraint.1

By organizing the problem into these matrix forms, mathematicians can apply systematic methods, such as the simplex method, to solve for the optimal values of $x$ efficiently.1