det

---
description: Guidelines for det
globs: **/*
---


 Usage

Calculates the determinant of a matrix. Note that for all
but very small problems, the determinant is not particularly
useful. The condition number cond gives a more reasonable
estimate as to the suitability of a matrix for inversion
than comparing det(A) to zero. In any case, the syntax for
its use is

    y = det(A)

where A is a square matrix.


 Internals

The determinant is calculated via the LU decomposition. Note
that the determinant of a product of matrices is the product
of the determinants. Then, we have that
 \[ L U = P A \]
where L is lower triangular with 1s on the main diagonal, U
is upper triangular, and P is a row-permutation matrix.
Taking the determinant of both sides yields
 \[ |L U| = |L| |U| = |U| = |P A| = |P| |A| \]
where we have used the fact that the determinant of L is 1.
The determinant of P (which is a row exchange matrix) is
either 1 or -1.


 Example

Here we assemble a random matrix and compute its determinant

  --> A = rand(5);
  --> det(A)

  ans =
     -0.0489

Then, we exchange two rows of A to demonstrate how the
determinant changes sign (but the magnitude is the same)

  --> B = A([2,1,3,4,5],:);
  --> det(B)

  ans =
      0.0489


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* Array_Generation_and_Manipulations
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