Matrix Determinants

[last updated: 2024-03-07]

Matrix home page
https://en.wikipedia.org/wiki/Determinant
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• The Determinant of a matrix
  is a number (a scaler) that is a property/parameter of the matrix.
  The Determinant of a matrix A is denoted as: det(A)
  Determinates are sometimes denoted as: |A|, however I use that symbol to represent the "absolute value" or magnitude of a vector...
  Note: Determinants can only be calculated for Square matrices
        ... but there is a workaround???

• Determinants can be calculated.
  Note: There is more than one way to calculate the determinant.

• Calculate the Determinant of a matrix using the Laplace Expansion method:
  • For a 2x2 matrix: A = k p and r s
    The determinant is: |A| = (k * s) - (p * r)
  • For a 3x3 matrix:
    a b c
    d e f
    g h i

    The determinant is:
    |A| = [a * (ei - fh)] - [b * (di - fg)] + [c * (dh - eg)]

    The pattern is:

    Multiply a by the determinant of the 2×2 matrix that is not in a's row or column.
    Repeat for b, and c
    Sum a & c calculation, but subtract the b
  • For 4x4 and higher dimension matrices:
    plus a times the determinant of the matrix that is not in a's row or column,
    minus b times the determinant of the matrix that is not in b's row or column,
    plus c times the determinant of the matrix that is not in c's row or column,
    minus d times the determinant of the matrix that is not in d's row or column,

• A square matrix with a Determinant = 0 does not have an inverse (is not invertible).
  It is called a Singular or Degenerate matrix.

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