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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 D1  
[C]: Coefficients; this is a square matrix, n x n, because there are the same number of equations as unknowns. 
The three matrices for equations 

are: 

To determine the values for the variables, the matrix expression must be solved for matrix [U]. The matrix algorithm to do so is:
[U]=[C^{T}C]^{1} x [C^{T}K] 
Equation D2 
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 [C^{T}] and [C]
(4) Determine the inverse of [C^{T}C]
Use Row Manipulation
(5) Multiply [C^{T}] and [K]
(6) Multiply [C^{T}C]^{1} and [C^{T}K] to get [U]
(7) Optional: check [K] = [C] x [U]
(8) Solution summary
x=2.791
y=1.199
z=2.086