Area of Parallelogram and Determinant of matrix, and linear transformation

 








above from ref.11

Below from ref. 12

A determinant is a property of a square matrix.

The value of the determinant has many implications for the matrix. A determinant of 0 implies that the matrix is singular, and thus not invertible. 

A system of linear equations can be solved by creating a matrix out of the coefficients and taking the determinant; this method is called Cramer's rule, and can only be used when the determinant is not equal to 0. Geometrically, the determinant represents the signed area of the parallelogram formed by the column vectors taken as Cartesian coordinates.

There are many methods used for computing the determinant. Some matrices, such as diagonal or triangular matrices, can have their determinants computed by taking the product of the elements on the main diagonal. 

For a 2-by-2 matrix, the determinant is calculated by subtracting the reverse diagonal from the main diagonal, which is known as the Leibniz formula. The determinant of the product of matrices is equal to the product of determinants of those matrices, so it may be beneficial to decompose a matrix into simpler matrices, calculate the individual determinants, then multiply the results. Some useful decomposition methods include QR, LU and Cholesky decomposition. For more complicated matrices, the Laplace formula (cofactor expansion), Gaussian elimination or other algorithms must be used to calculate the determinant.


Reference

  1. https://www.mathsisfun.com/geometry/parallelogram.html
  2. https://mathworld.wolfram.com/Parallelogram.html
  3. https://www.mathplanet.com/education/geometry/quadrilaterals/properties-of-parallelograms
  4. https://math.stackexchange.com/questions/29128/why-determinant-of-a-2-by-2-matrix-is-the-area-of-a-parallelogram
  5. determinant as the area (in 2D; in 3D it would be the volume, etc.) of the parallelogram made by the vectors, https://math.stackexchange.com/questions/668/whats-an-intuitive-way-to-think-about-the-determinant
  6. https://www.geogebra.org/m/Un6WZ3gs Developing the concept of a determinant using DGS
  7. What really is matrix determinant? https://towardsdatascience.com/what-really-is-a-matrix-determinant-89c09884164c
  8. https://www.statlect.com/matrix-algebra/determinant-of-a-matrix
  9. https://towardsdatascience.com/discovering-the-matrix-determinant-11b56616d99a
  10. https://mathinsight.org/determinant_linear_transformation
  11. What Is Determinant (cut-the-knot.org)
  12. Determinant Calculator: Wolfram|Alpha (wolframalpha.com)



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