Matrices and Simultaneous Equations

Matrices and Simultaneous Equations

Basics

  • A matrix is a rectangular array of numbers arranged in rows and columns.
  • Matrices can be used to solve systems of simultaneous equations.
  • Each row in a matrix corresponds to an equation in the system, and each column corresponds to the coefficient of a variable.

Solving Simultaneous Equations using Matrices

  • A simultaneous system of equations can be represented in the form of AX=B, where A is the matrix of coefficients, X is the column matrix of variables and B is the column matrix of constants.
  • To solve for X, the inverse of the coefficient matrix A is used, which yields X = A^(-1)B.
  • This method can only be used if the matrix A is invertible, i.e. its determinant is not zero. If the determinant of A is zero, the system of equations has either infinitely many solutions or no solutions.

Row Operations

  • An alternative approach is to use row operations to simplify the augmented matrix [A B] to its reduced row-echelon form (RREF).
  • There are three types of row operations: swapping two rows, multiplying a row by a non-zero scalar, and adding a multiple of one row to another.
  • The RREF provides the solution to the simultaneous equations, with each row corresponding to an equation and each column to a coefficient of a variable.

Interpreting the Results

  • If an all-zero row in the RREF corresponds to a non-zero constant in the B matrix, the system is inconsistent and has no solutions.
  • If there are more variables than equations (rows), then the system is said to be underdetermined, and may have infinitely many solutions.
  • If the determinant of the coefficient matrix is not zero, then the system has a unique solution.

Applications

  • Solving simultaneous equations using matrices is useful in many fields, such as physics, engineering and economics, where multiple variables are interconnected.
  • They are particularly useful in linear programming and game theory, where the goal is often to find the optimal solution under given constraints.