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 \( U \):
- If \( B = CD \) then \( \text{det}(B) = \text{det}(C) \text{det}(D) \; \).
- The determinant of a triangular matrix is the product of its diagonal elements.
- 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 \( \text{det}(A) = (4)(1)(-8) = -32 \).