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Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.3: Symmetric Matrices And Orthogonal Diagonalization

Problem 57E

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Proof Prove that if A is an orthogonal matrix, then so are AT and A-1.

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Proof Prove that if matrix A is diagonalizable, then AT is diagonalizable.
Proof Let A be an nn square matrix. Prove that the row vectors of A are linearly dependent if and only if the column vectors of A are linearly dependent.
Proof Prove that if A is an nn matrix, then A-AT is skew-symmetric.
Proof Prove that if A is an nn matrix that is idempotent and invertible, then A=In.
Proof Let A be a nonsingular matrix. Prove that if B is row-equivalent to A, then B is also nonsingular.
Proof Prove that the main diagonal of a skew-symmetric matrix consists entirely of zeros.
Proof Prove that if A and B are similar matrices and A is nonsingular, then B is also nonsingular and A1 and B1 are similar matrices.
Proof Let A be a nonsingular matrix of order 3. Prove that if {v1,v2,v3} is a linearly independent set in M3,1, then the set {Av1,Av2,Av3} is also linearly independent. Explain, by means of an example, why this is not true when A is singular.
Proof Prove that if A and B are square matrices of order n, then TrAB=TrBA.
Proof Let A and B be nn matrices such that AB=I.Prove that |A|0 and |B|0.
Proof Prove that each statement is true when A and B are square matrices of order n and c is a scalar. a TrA+B=TrA+TrB b TrcA=cTrA
Proof Prove that if A and B are nn skew-symmetric matrices, then A+B is skew-symmetric.

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