To prove that the determinant of the Vandermonde matrix V is
Explanation of Solution
Given information:
Vandermonde matrix V is given by
Explanation:
Start with induction on n .
For n = 1, the Vandermonde matrix is of size
Now, assume that the given condition is true for
Start from second last column on right side that is
The determinant will not be changed from this operation. All entries in the top row, except the leftmost one, have become zero as
After taking factors out,a
Thus, this completes the induction.
Want to see more full solutions like this?
- The meet of two zero-one matrices A and B is described as AAB = [ajj A bj] AvB = [aj A bijl] A v B = [aj v bijl A AB = [aj v bijl]arrow_forwardConsider the following. -4 2 0 1 -3 A = 0 4 2 2 1 2 2 (a) Verify that A is diagonalizable by computing P AP. P-lAP = E (b) Use the result of part (a) and the theorem below to find the eigenvalues of A. Similar Matrices Have the Same Eigenvalues If A and B are similar n xn matrices, then they have the same eigenvalues. VAarrow_forwardWhat is the sum of matrices M=(1,0,3)(0,-3,1) and N=(0,1,2)(0,0,-3)?arrow_forward
- Please solve for d, e, and farrow_forwardWhat is the product of matrices M=(-1,0)(0,3) and N=(0,1)(0,-3)?arrow_forwardPerform the following Matrix Operations for the predefined matrices. Given the System of equations: 2х + 4y — 5z + Зw %3D —33 3х + 5у—2z + бw %3D — 37 х — 2у + 4z — 2w 3 25 Зх + 5у-3z + Зw = -28 Write the systems as Ax = b, where A is the coefficient matrix and b is the vector for the constants. 1. Encode the Matrix A and the column vector b. 2. Solve for Determinant of A. 3. Find the Inverse of A. 4. Form the Reduced Row Echelon of A. 5. Find the number of rows and number of columns of Ab. 6. Find the sum of the columns of A. 7. In each of the columns of A, find the highest values and its indices. 8. Augment A with b; 9. Find b\A 10. Form the Reduced Row Echelon of Ab. 11. Extract the Last Column of the Reduced Row Echelon Form of Ab. 12. Create a matrix A whose elements are the same as matrix A, but the first column is the column vector b. 13. Create a matrix A whose elements are the same as matrix A, but the second column is the column vector b. 14. Create a matrix A whose elements…arrow_forward
- Operations Research : Applications and AlgorithmsComputer ScienceISBN:9780534380588Author:Wayne L. WinstonPublisher:Brooks Cole