Question

Use LU decomposition to determine the matrix inverse for the following system. Do not use a pivoting strategy, and check your results by verifying that [A][A]⁻¹ = [I].

10x₁+ 2x₂- x₃= 27
-3x₁- 6x₂+2x₃= -61.5
x₁+ x₂+ 5x₃=-21.5

Answer 1

LU decomposition is used to find the inverse of a matrix, but the given question lacks a complete matrix or system of equations. Correct application of this method involves finding L and U matrices before solving for the inverse, and checking that the product of A and its inverse equates to the identity matrix.

Step-by-step explanation:

The question involves using LU decomposition to find the inverse of a matrix derived from a system of linear equations. LU decomposition is a method where a matrix A is decomposed into two matrices, L (lower triangular matrix) and U (upper triangular matrix), such that A = LU. The inverse of A, if it exists, can then be determined using these triangular matrices. However, the matrix or system of equations provided in the question seems to have some missing or unclear information. To correctly use LU decomposition, a complete matrix A is needed, and the steps would involve decomposing A into L and U, then solving for LY = I to find Y, and subsequently UX = Y to find X which would be A-1. The final step is to confirm that the multiplication of A and A-1 yields the identity matrix I.