Python code to calculate the eigenvalues and eigenvectors of a matrix from the numpy linalg library is eig. It returns the eigenvalues and the eigenvectors as one eigenvector per column of a matrix as shown in the code below. The vector w contains the eigenvalues of the matrix A and the matrix V has one eigenvector per column. Eigenvalues for the given matrix A are 2, 1 and 4 and we can see that determinant of A is equal to the product of these values and the trace of A to the sum.

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Randomized Select Algorithm

A randomized algorithm is defined as one that uses a certain amount of randomization in its logic or operation. It is said to be an algorithm that depends on the random number to perform its operation. It is used for reducing the overall time complexity, space complexity in the standard algorithm. The randomized algorithm uses the concept of random numbers to find the solutions to the different kinds of problems and also to improve the problem solutions.