Matrix decompositions, like LU decomposition and QR decomposition, play an important role in calculating determinants, which are special numbers that show certain properties of matrices. Here’s how these decompositions affect determinants:

1. LU Decomposition:

   • For a square matrix A, the LU decomposition breaks A down into two simpler matrices: a lower triangular matrix L and an upper triangular matrix U.
   • This can be shown as: $A = LU$
   • To find the determinant of A, you can use: $\det(A) = \det(L) \cdot \det(U)$
   • Since the determinant of a triangular matrix is just the product of its diagonal elements, if L has ones on its diagonal, you can simplify this to: $\det(A) = \prod_{i=1}^{n} u_{ii}$

2. QR Decomposition:

   • In QR decomposition, A is broken down into a matrix Q that is orthogonal and an upper triangular matrix R. It can be represented as: $A = QR$
   • The determinant in this case is affected by the orthogonal matrix: $\det(A) = \det(Q) \cdot \det(R)$
   • Since the determinant of Q is either +1 or -1, it simplifies to: $\det(A) = \det(R)$

3. Impact on Speed of Calculations:

   • Matrix decompositions make it easier to calculate determinants for large matrices. Finding determinants directly can take a lot of time and effort, with a complexity of $O(n!)$.
   • But with LU or QR decomposition, it’s much faster, with a complexity of just $O(n^3$. This makes it easier to use in real-life applications.

In short, matrix decompositions not only help us calculate determinants more easily but also help us understand the structure and properties of the matrices better.