Prove that a permutation matrix is orthogonal.

Q: Show that an n x n nonnegative matrix is transitive if and only if £=1 4* is a positive matrix. -k=1

A: Given that, An nonnegative matrix is transitive.…

Q: Diagonalize matrix A, i.e., find diagonal D and invertible P such that A = PDP-1, where -[:] 3 2 A =…

A:

Q: The nth order permutation matrix refers to an n × n matrix in which each column and row has exactly…

A: The series expansion of exponential function is given by ex=1+x1!+x22!+x33!+... This series also…

Q: Prove Any n by n triangular matrix with main diagonal entriesare all 1 is diagonalizable if and only…

A: Triangular Matrix: A triangular matrix is a square matrix where all the values above or below the…

Q: Let {v₁,...Vn} R" be linearly independnt. (a) Let A be a singular n × n matrix. Prove or disprove…

A: Let a set of vectors be For m scalars  satisfyIf the only solution is Then the vectors  are linearly…

Q: Find a matrix P such that PTAP orthogonally diagonalizes A. Verify that PTAP gives the correct…

A: The given matrix is A=1100110000110011 We have to find matrix P. Also, we have to find PTAP is…

Q: Explain the fact “Every permutation matrix is an orthogonal matrix”.

A: A square matrix A is orthogonal if AT=A-1. A permutation matrix is the result of repeatedly…

Q: 5. Give 2 x 2 matrices A so that for any x e R² we have, respectively, (f) Axis the vector obtained…

A:

Q: Given a particular matrix A and a vector x + 0, discuss how to determine whether x is an eigenvector…

A: Let A is any matrix of order n×n. The scalar λ is an eigen value of A when there is non zero vector…

Q: Prove that a 2×2 matrix transforms any straight line into a second straight line and points on the…

A: Let T be a 2x2 matrix given as T = abcd.Let the line be given by y = mx+c.Any point on the line can…

Q: Prove that the identity matrix cannot be the product of three row exchanges ( or five). It can be…

A: An identity matrix is a square matrix of any order which contains on its main diagonal elements with…

Q: TRUE/FALSE Assume that all the given matrices are n × n. If A = O, but Ak = O (where O denotes the…

A: Given:   To Check: A is singular or not Concept: A is singular only when det(A) = 0

Q: Prove that every symmetric upper (or lower) triangular matrix is diagonal.

A: Solution: Consider the given statement " Every symmetric upper (lower) triangular matrix is…

Q: Prove that there is no 2 x 5 standard generator matrix G that willproduce a (5, 2) linear code over…

A: Let C be a 5,2 linear code over Z3 such that it can detect all possible triple errors. Then the…

Q: Show that if A is a reversible matrix on an F body, the At transpose is also reversible.

A: Given that A is a reversible matrix. Then, AA-1=I, where I is an identity matrix and A-1 is the…

Q: Define the term permutation matrix.

A:

Q: Show that if P is a symmetric permutation matrix then P2k = I and P2k+1 = P.

A: P is a symmetric permutation Matrix.One has to prove that .

Q: Question : Assume that T is a linear transformation. Find the standard matrix of T OT:R² → R² first…

A:

Q: a) Decompose the coefficient matrix A into lower and upper triangular matrices.

A:

Q: Is the following matrix 'B' diagonalizable?

A:

Q: 1 1 -2 -4 A . 1 -4 and -4 4 A 1 A = ||

A: We have to solve given problem:

Q: Prove that the nonzero row vectors ofa matrix in row-echelon form are linearly independent. Let = A…

A:

Q: Show that if A is an n × n antisymmetric square matrix with n odd, then A is not invertible.

A:

Q: Prove that if the diagonal entries of a diagonal matrix are permuted, then the resulting diagonal…

A: Permuting the diagonal entries of a diagonal matrix D amounts to permuting the rows of that matrix…

Q: Question : Assume that T is a linear transformation. Find the standard matrix of T O T:R² → R³,…

A: Any linear transformation on finite dimensional spaces can be written as a matrix format.

Q: Prove that for a diagonal matrix, the 1-,2-,∞- and -norms are the same.

A: We are given a diagonal matrix D. We wish to show that D1=D2=D∞ If we write D=d10⋯00d2⋱⋮⋮⋱⋱00⋯dn…

Q: Show that for a square matrix A, AAT is a symmetric matrix.

A: