Introduction to Algorithms
3rd Edition
ISBN: 9780262033848
Author: Thomas H. Cormen, Ronald L. Rivest, Charles E. Leiserson, Clifford Stein
Publisher: MIT Press
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Chapter D.1, Problem 2E
Program Plan Intro
To prove:
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. The determinant of an n X n matrix can be used
in solving systems of linear equations, as well
as for other purposes. The determinant of A can
be defined in terms of minors and cofactors. The
minor of element aj is the determinant of the
(n – 1) X (n – 1) matrix obtained from A by
crossing out the elements in row i and column j;
denote this minor by Mj. The cofactor of element
aj, denoted by Cj. is defined by
Cy = (-1y**Mg
The determinant of A is computed by multiplying
all the elements in some fixed row of A by their
respective cofactors and summing the results. For
example, if the first row is used, then the determi-
nant of A is given by
Σ (α(CI)
k=1
Write a program that, when given n and the entries
in an n Xn array A as input, computes the deter-
minant of A. Use a recursive algorithm.
Find the eigenvalues of the matrix and determine whether there is a sufficient number to guarantee that the matrix is diagonalizable. (Recall that the matrix may be diagonalizable even though it is not guaranteed to be diagonalizable by the theorem shown below.)
Sufficient Condition for Diagonalization
If an n xn matrix A has n distinct eigenvalues, then the corresponding eigenvectors are linearly independent and A is diagonalizable.
Find the eigenvalues. (Enter your answers as a comma-separated list.)
Is there a sufficient number to guarantee that the matrix is diagonalizable?
O Yes
O No
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Assume A is k x n-matrix and P is k × k-invertible matrix. Prove that rank(PA) = rank(A).
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- Find the value of 'a', if B is a singular matrix? %3D (A) 5 9 (a) (C) 7arrow_forwardGive atleast 3 examples of each type of matrix. I. Idempotent Matrixarrow_forwardFor the matrix A, find (if possible) a nonsingular matrix P such that p-lAP is diagonal. (If not possible, enter IMPOSSIBLE.) 6 -3 A = -2 P = Verify that P1AP is a diagonal matrix with the eigenvalues on the main diagonal. p-1AP =arrow_forward
- Using Python, to help solve the following problem. Provide an explanation of your solutions to the problem. 4. A symmetric matrix D is positive definite if x¹TDx > 0 for any nonzero vector x. It can be proved that any symmetric, positive definite matrix D can be factored in the form D = LLT for some lower triangular matrix L with nonzero diagonal elements. This is called the Cholesky factorization of D. Consider the matrix [2.25 -3 4.5 -10 -3 5 4.5 -10 34 a. Is A positive definite? Explain. A = b. Find a lower triangular matrix L such that LLT = A.arrow_forwardList any two properties of eigenvalues of a square matrix. Explain it in your own way and provide it with examples.arrow_forwardFind the determinant of the matrix |4 0 -7 3. -5| ... 2 3. -6 4 -8 5 0 2. -3 -1 A) 12 B © -6 D) -12arrow_forward
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