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Math and science::Algebra

Determinant from pivots

Let \( A \) be a matrix that is decomposed through elimination to \( A = LU \). The truth of the following statements allow us to read the determinant of \( A \) from the pivots that are placed along the diagonal of \( U \):

  1. If \( B = CD \) then \( \text{det}(B) = \text{det}(C) \text{det}(D) \; \).
  2. The determinant of a triangular matrix is the product of its diagonal elements.
  3. All entries along the diagonal of \( L \) are 1.

From these statements, we can see that the determinant of \( A \) is the product of the pivots that are placed along the diagonal of \( U \).


Example

\[ A = \begin{bmatrix} 4 & 2 & 0 \\ 2 & 2 & 3 \\ 0 & 3 & 1 \end{bmatrix} \]

Which is decomposed as \( A = LU \) to:

\[ A = LU = \begin{bmatrix} 1 & 0 & 0 \\ \frac{1}{2} & 1 & 0 \\ 0 & 3 & 1 \end{bmatrix} \begin{bmatrix} 4 & 2 & 0 \\ 0 & 1 & 3 \\ 0 & 0 & -8 \end{bmatrix} \]

The determinant of \( A \) is the product of the pivots along the diagonal of \( U \), so \( \text{det}(A) = (4)(1)(-8) = -32 \).