Let A be a symmetric matrix having only one eigenvalue à and C be a matrix that diagonalizes A by a similarity transformation. Find a simplified expression for A in terms of 2, C, and I, the identity matrix.

Q: If X is a rankr matrix, show that the two square matrices XXH and X#X have the same nonzero…

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Q: Find a 2×2 matrix A such that [0                    [-3 1]       and        -2] are eigenvectors…

A: On solving this we will get-:

Q: Find all eigenvalues and the basis of each eigenspace for the matrix. -4 -2) A= -3 4 -3 1 If A is…

A: Given a matrix A. We need to find the eigenvalues and the basis for each eigenspace. Also if A is…

Q: If 2 is one of the eigen vector of a matrix A = |1 ] A3x3 has three distinct eigen values then which…

A: INTRODUCTION: In linear algebra, eigenvalues are related to eigenvectors. In the analysis of linear…

Q: Let A be an nxn matrix. Prove that if A is orthogonally diagonalizable (and has real eigenvalues),…

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Q: Suppose A is a diagonalizable matrix and the only eigenvalues of A are 0 and 1. Prove that A is…

A: Given A is a diagonalizable matrix and the only eigenvalues of A are 0 and 1. To  prove A is…

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A: The given matrix A is 5x5 matrix. One of the eigenvalue has a three-dimensional eigenspace, i.e. it…

Q: . Let 2 and u are distinct eigenvalues of matrix A, let x be a right eigenvector for 2, and let y be…

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Q: Give an example of an m x n matrix A such that the nonzero eigenvalues of A'A are distinct from…

A: We have to solve given problem:

Q: Consider the matrix M Establish %3D 3 for which value of k, X 3 is an eigenvalue of M. In this case,…

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Q: Let A be a 5×5 matrix with eigenvalues: 0, 2, 3, -1, and 6. Then the matrix A is* not diagonalizable…

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Q: If A is 4 x 4 matrix with 4 linearly independent eigenvectors then A is diagonalizable. Select one:…

A: yes this is the fact If any matrix have same number of eigenvalues as his order then it is called…

Q: If A and B are nxn matrices and v is an eigenvector of A with corresponding eigenvalue A and v is…

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Q: Let A be a 3 x 3 diagonalizable matrix with eigenvalues: A, = 6 with %3D corresponding eigenvector…

A: It is given that the eigenvalues of the matrix A of order 3 are λ1=6, λ2=2 and  λ3=1. The…

Q: If A, B are nxn matrices, suppose A and B have the same eigenvalues A,..., withn linearly…

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Q: Let A be a 3 x 3 diagonalizable matrix whose eigenvalues are X, = -3, A2 2, and Ag=-2 with…

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Q: Construct a symmetric matrix A with eigenvalues A1 = -3 and A2 = 10 and corresponding orthogonal…

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Q: Show that the eigenvalues of the upper triangular matrix: A = (* ) a b' 0 d) are A= a and X = d.…

A: here we find the eigen values and eigenspaces of the triangular matrix. ab0d

Q: Prove that if A is a 3 x 3 non-invertible matrix possessing a negative eigenvalue and with positive…

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Q: Let A be a diagonalizable matrix whose eigenvalues are all either 1 or −1. Show that A−1 = A.

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Q: Show that a matrix of the same size but with no eigenvalues that are real is similar to the matrix:…

A: Let us consider a n×n matrix. we know that Characteristic polynomial of an n×n matrix A is degree n…

Q: Show that is an eigenvector of the matrix and find the corresponding eigenvalue.

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Q: Let D be a 3 x 3 diagonal matrix with diagonal entries A1,...,A3, which are distinct real numbers.…

A: Let D be a 3×3 diagonal matrix with diagonal entries λ1, λ2, and λ3, are the distinct real numbers.…

Q: Find a 2 x 2 matrix A for which E4 = span {[3]}, where Ex denotes the eigenspace associated with the…

A: We will find the matrix A using the given information.

Q: If A is 4 x4 matrix with 4 linearly independent eigenvectors then A is diagonalizable. Select one: O…

A: If A is 4 x 4 matrix with 4 linearly independent eigenvectors  To determine A is diagonalizable

Q: Let A be a Hermitian matrix. Show that all its eigenvalues ​​belong to the real axis.

A: See proof below

Q: Let A and B be n x n matrices with A invertible. Prove that AB and BA have the same eigenvalues.

A: The two matrices AB and BA will have the same Eigen values if they have the same characteristic…

Q: Suppose that A is a square matrix in which the sum of the entries of each row equals the same scalar…

A: Suppose that A is a square matrix in which the sum of the entries of each row equals the same scalar…

Q: Let A be an n×n matrix and let λ be an eigenvalue of A. If A−λI has rank k, what is the dimension of…

A: Given : A is a n×n matrix with λ as it's eigen value  Let B= A-λI be another matrix with k as it's…

Q: Suppose B is a diagonalizable matrix and the only eigenvalues of B are −1 and 1. Prove that B is…

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Q: 2. Determine whether there exists a 3 x 3 symmetric matrix whose eigenvalues are A1 = -1, A2=3, A3 =…

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Q: -2 3.

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Q: For matrices A and B with B invertible AB and BA have the same eigenvalues.

A: soln:If A & B are two square matrices of order n then, AB & BA have same eigen values.Let A…

Q: Let I be the 5 × 5 identity matrix. What is the algebraic multiplicity of its eigenvalue? Answer:

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Q: Let A be an anti-Hermitian matrix. Show that all its eigenvalues ​​belong to the imaginary axis.

A: We must know that all the eigen values of a hermitian matrix are always real.

Q: Is the following matrix diagonalizable? If so, provide the matrices P and D. If not, explain why not…

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Q: Find the matrix, A, with eigenvectors and whose eigenvalues are 2 and 3 respectively. A =

A: Given that,  Eigen values are: λ1=2 and λ2=3 Eigen vectors: 01,-1-2 Let M be the matrix defined by…

Q: Calculate whether the given matrices are diagonalizable or not. That is, for each matrix, calculate…

A: We will use the fact if the eugen values of a matrix are distinct then the matrix is diagonalizable.

Q: If A be nxn matrix with eigen value L then find the eigen value of inverse of A .

A: The given matrix A be an n×n matrix with eigenvalue L. Calculate the eigen value of inverse of A.

Q: Suppose that A is an n×n matrix with n eigenvalues μ1, μ2, ..., μn. Show that the determinant of A…

A: Consider the provided question, We have to show that determinant of matrix A is equal to the product…

Q: Let A be an invertible matrix with eigenvalue λ. Show that 1/λ is an eigenvalue of A−1.

A: Given : A is an invertible matrix with eigenvalue λ. By definition we have, Ax = λx , λ ≠ 0

Q: Let A be a matrix whose columns all add up to a fixed constant δ. Show that δ is an eigenvalue of A.

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Q: Let A be a symmetric n x n matrix. Let A1 and A2 be two eigenvalues with eigenvectors vị and v2 for…

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Q: Let A and B be similar matrices. Prove that the geometric multiplicities of the eigenvalues of A and…

A: Given: A and B be similar matrices. To prove: The geometric multiplicities of the eigenvalues of A…

Q: Let A be the 2 x 2 matrix with elements a,, a12 = a21 = + 1 and a,, = - 1. Then the %3D eigenvalues…

A: Introduction :Given , a matrix A with elements and we have asked for the eigen values of matrix A19

Q: Let A be the matrix A = ( 1 -2 -1 -6 0 a) Find all eigenvalue and coresponding eigenvector of A;

A: Explanation of the solution is given below...

Q: Let A be a 5×5 matrix with eigenvalues. 1,0. 3. -1, and -6. Then the matrix A is * not…

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Q: If B and C are positive matrices, show that A has a positive real eigenvalue r with the property…

A: We have to use the properties and definitions of eigenvalues and eigenvectors of the matrix and…