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8: Systems of Equations and Matrices

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    • 8.1: Systems of Linear Equations- Gaussian Elimination
      Up until now, when we concerned ourselves with solving different types of equations there was only one equation to solve at a time. Given an equation 𝑓(𝑥)=𝑔(𝑥) f ( x ) = g ( x ) , we could check our solutions geometrically by finding where the graphs of 𝑦=𝑓(𝑥) y = f ( x ) and 𝑦=𝑔(𝑥) y = g ( x ) intersect. The 𝑥 x -coordinates of these intersection points correspond to the solutions to the equation 𝑓(𝑥)=𝑔(𝑥) f ( x ) = g ( x ) , and the 𝑦 y -coordinates were largely ignored.
    • 8.2: Systems of Linear Equations- Augmented Matrices
      We previously introduced Gaussian Elimination as a means of transforming a system of linear equations into triangular form with the ultimate goal of producing an equivalent system of linear equations which is easier to solve. If we study the process, we see that all of our moves are determined entirely by the coefficients of the variables involved, and not the variables themselves. In this section, we introduce a bookkeeping device to help us solve systems of linear equations.
    • 8.3: Matrix Arithmetic
      Previously, we used a special class of matrices, the augmented matrices, to assist us in solving systems of linear equations. In this section, we study matrices as mathematical objects of their own accord, temporarily divorced from systems of linear equations.
    • 8.4: Systems of Linear Equations: Matrix Inverses
      We previously showed how we can rewrite a system of linear equations as the matrix equation AX=B where A and B are known matrices and the solution matrix X of the equation corresponds to the solution of the system. In this section, we develop the method for solving such an equation.
    • 8.5: Determinants and Cramer’s Rule
      n this section we assign to each square matrix A a real number, called the determinant of A, which will eventually lead us to yet another technique for solving consistent independent systems of linear equations. The determinant is defined recursively.
    • 8.6: Partial Fraction Decomposition
      This section uses systems of linear equations to rewrite rational functions in a form more palatable. This Modules demonstrated how rational functions cab be resolved into partial fractions.
    • 8.7: Systems of Non-Linear Equations and Inequalities
      In this section, we study systems of non-linear equations and inequalities. Unlike the systems of linear equations for which we have developed several algorithmic solution techniques, there is no general algorithm to solve systems of non-linear equations. Moreover, all of the usual hazards of non-linear equations like extraneous solutions and unusual function domains are once again present. Along with the tried and true techniques of substitution and elimination, we shall often need equal parts

    This page titled 8: Systems of Equations and Matrices is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by Carl Stitz & Jeff Zeager via source content that was edited to the style and standards of the LibreTexts platform.