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Published on May 29, 2025
Find the general solution to a system of differential equations X' = AX, given the eigenvalues and eigenvectors of the matrix A. The solution involves finding linearly independent solutions based on the eigenvalues and eigenvectors, including complex conjugate pairs.
Solve the initial value problem (IVP) for a system of linear differential equations X' = AX, finding the particular solution given the matrix A and initial condition X(0). The solution involves finding eigenvalues, eigenvectors, and applying the initial condition.
The provided text details the process of performing LU decomposition on a 4x4 matrix, including row swaps and elementary row operations to achieve an upper triangular matrix (U) and a lower triangular matrix (L). It also calculates the corresponding diagonal matrix (D) and permutation matrix (P) to complete the decomposition. The solution demonstrates the Gaussian elimination method to find the inverse of a 3x3 matrix.
The eigenvalues of the matrix [[3, 1], [1, 3]] are 4 and 2.
The eigenvalues of the matrix [[0, 1], [-1, 0]] are i and -i.
The eigenvalues of the given 3x3 matrix are 1, 2, and 2.
The determinant of the given 4x4 matrix is calculated using cofactor expansion along the fourth row, resulting in a determinant of -21.
The determinant of the 2x2 matrix [[2, 2], [3, 5]] is calculated as (2 * 5) - (2 * 3) = 4.
The determinant of the 3x3 matrix is calculated using the formula a(ei - fh) - b(di - fg) + c(dh - eg), resulting in a determinant of 0.