Matrices to Solve Linear Equations

[last updated: 2023-01-17]

Unfinished!

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• For example, suppose you have two linear equations,
  that is, there are no exponents on the variables.
  Each equation will describe a straight line in a 2-dimensional plane.
  x -2y = 0
  x -y = 1

  
  To solve this set/system, you find the point in 2-d space where the lines intersect.

  • The set/system of these two equations would be represented by these 3 matrices:
    • this coefficient matrix (we'll call it A):
      1 -2
      1 -1
      where each number in the matrix represents a coefficient of a variable
    • this variable matrix (we'll call it V):
      x
      y
    • and this constant matrix (we'll call it C):
      0
      1
    • The system is then represented by:
      A V = C

• To solve this system of equations, ie. find the point where they intersect, multiply both sides by the inverse of A:
  V = (A^-1) C
  However: The coefficient matrix A may not be invertible.
  This can happen if there are no solutions, or infinitely many solutions, or the system of equations is inconsistent/contradictory.
  If the matrix is not invertible, then the determinate will be 0.

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