D. Equations and Matrices

1. Linear Equations

a. Matrix Algorithms

A set of simultaneous linear equations can be denoted in matrix form as:

[C] x [U] = [K]

Equation D-1

[C]: Coefficients; this is a square matrix, n x n, because there are the same number of equations as unknowns.
[U]: Variables; n x 1
[K]: Constants; n x 1

The three matrices for equations




To determine the values for the variables, the matrix expression must be solved for matrix [U]. The matrix algorithm to do so is:

[U]=[CTC]-1 x [CTK]

Equation D-2

b. Example

Given the following equations, solve for x, y, and z using matrices.

(1) Set up [C], [U], and [K] matrices


(2) Transpose [C]


(3) Multiply [CT] and [C]


(4) Determine the inverse of [CTC]

Use Row Manipulation


(5) Multiply [CT] and [K]


(6) Multiply [CTC]-1 and [CTK] to get [U]


(7) Optional: check [K] = [C] x [U]


(8) Solution summary