Solving Systems of Linear Equations with Matrices

Systems of Linear Equations

A system of linear equations can be written in matrix form as Ax = b, where A is the coefficient matrix, x is the vector of unknowns, and b is the constants vector.

Example

2x + 3y = 8      | 2  3 | | x |   | 8 |
4x + y  = 6  =>  | 4  1 | | y | = | 6 |

Method 1: Direct Inverse

If A is invertible: x = A^(-1) * b

det(A) = 2 - 12 = -10

A^(-1) = (-1/10) * | 1  -3 | = | -0.1   0.3 |
                    | -4   2 |   |  0.4  -0.2 |

x = | -0.1   0.3 | | 8 | = | -0.8 + 1.8 | = | 1 |
    |  0.4  -0.2 | | 6 |   |  3.2 - 1.2 |   | 2 |

So x = 1, y = 2.

Method 2: Gaussian Elimination

Form the augmented matrix [A|b] and reduce to REF, then back-substitute.

Method 3: Cramer's Rule

For small systems, each variable can be found by replacing the corresponding column of A with b and computing the ratio of determinants:

x_i = det(A_i) / det(A)

Solution Types

• Unique solution: rank(A) = rank([A|b]) = n (number of unknowns)
• No solution: rank(A) < rank([A|b]) (inconsistent system)
• Infinite solutions: rank(A) = rank([A|b]) < n (underdetermined)