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demipaul
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How would I calculate the determinant of a square matrix using LU Decomposition. Please be plain, I am not good with technical terms. An example would be nice. Thank you!
(Note: A http://planetmath.org/encyclopedia/UnitLowerTriangularMatrix.html" is a triangular matrix with 1's along it's diagonals)The matrices L and U can be used to compute the determinant of the matrix A very quickly, because det(A) = det(L) det(U) and the determinant of a triangular matrix is simply the product of its diagonal entries. In particular, if L is a unit triangular matrix, then
[tex]det(A) = det(L)det(U) = 1 \cdot det(U) =\prod^{n}_{i = 1}u_{ii}.[/tex]
LU decomposition is a method used to factor a square matrix into a lower triangular matrix (L) and an upper triangular matrix (U). It is important because it allows for more efficient calculations in solving systems of linear equations and finding the determinant of a matrix.
The process involves first factoring the original matrix into L and U matrices. Then, the determinant of the original matrix can be calculated by multiplying the determinants of the L and U matrices together. The determinant of a triangular matrix is simply the product of its diagonal elements.
No, LU decomposition can only be used for square matrices. For non-square matrices, other methods such as QR decomposition or singular value decomposition may be used.
It depends on the size and structure of the matrix. LU decomposition is more efficient for larger matrices, but for smaller matrices, other methods may be more efficient.
One limitation is that LU decomposition can only be used for invertible matrices, meaning those with a non-zero determinant. It also requires more computation compared to other methods such as using the cofactor expansion formula.