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All Textbook Solutions for Elementary Linear Algebra (MindTap Course List)
55E56E57E58E59E60E61E62E63E64E65E66E67EUse your schools library, the Internet, or some other reference source to find the real-life applications of constrained optimization.Characteristic Equation, Eigenvalues, and Basis In Exercises 1-6, find a the characteristic equation of A, b the eigenvalues of A, and c a basis for the eigenspace corresponding to each eigenvalue. A=[2152]Characteristic Equation, Eigenvalues, and Basis In Exercises 1-6, find a the characteristic equation of A, b the eigenvalues of A, and c a basis for the eigenspace corresponding to each eigenvalue. A=[2142]Characteristic Equation, Eigenvalues, and Basis In Exercises 1-6, find a the characteristic equation of A, b the eigenvalues of A, and c a basis for the eigenspace corresponding to each eigenvalue. A=[9432061411]4CRCharacteristic Equation, Eigenvalues, and Basis In Exercises 1-6, find a the characteristic equation of A, b the eigenvalues of A, and c a basis for the eigenspace corresponding to each eigenvalue. A=[201034001]6CRCharacteristics Equation, Eigenvalues, and Basis In Exercises 7 and 8, use a software program or a graphing utility to find a the characteristics equation of A, b the eigenvalues of A, and c a basis for the eigenspace corresponding to each eigenvalue. A=[2100120000210012]Characteristics Equation, Eigenvalues, and Basis In Exercises 7 and 8, use a software program or a graphing utility to find a the characteristics equation of A, b the eigenvalues of A, and c a basis for the eigenspace corresponding to each eigenvalue. A=[3020131001100004]Determining Whether a Matrix Is DiagonalizableIn Exercises 9-14, determine whether A is diagonalizable. If it is, find a nonsingular matrix P such that P-1AP is diagonal. A=[1428]10CRDetermining Whether a Matrix Is DiagonalizableIn Exercises 9-14, determine whether A is diagonalizable. If it is, find a nonsingular matrix P such that P-1AP is diagonal. A=[213012001]12CRDetermining Whether a Matrix Is DiagonalizableIn Exercises 9-14, determine whether A is diagonalizable. If it is, find a nonsingular matrix P such that P-1AP is diagonal. A=[102010201]14CRFor what values of a does the matrix A=[01a1] have the characteristics below? a A has eigenvalue of multiplicity 2. b A has 1 and 2 as eigenvalues. c A has real eigenvalues.16CRWriting In Exercises 17-20, explain why the given matrix is not diagonalizable. A=[0900]18CRWriting In Exercises 17-20, explain why the given matrix is not diagonalizable. A=[300130003]20CRDetermine Whether Two Matrices Are Similar In Exercises 21-24, determine whether the matrices are similar. If they are, find a matrix P such that A=P1BP. A=[1002],B=[2001]Determine Whether Two Matrices Are Similar In Exercises 21-24, determine whether the matrices are similar. If they are, find a matrix P such that A=P1BP. A=[5003],B=[7241]Determine Whether Two Matrices Are Similar In Exercises 21-24, determine whether the matrices are similar. If they are, find a matrix P such that A=P1BP. A=[110011001],B=[110010001]Determine Whether Two Matrices Are Similar In Exercises 21-24, determine whether the matrices are similar. If they are, find a matrix P such that A=P1BP. A=[100020002],B=[133353331]Determining Symmetric and Orthogonal Matrices In Exercises 25-32, determine whether the matrix is symmetric, orthogonal, both or neither. A=[22222222]26CRDetermining Symmetric and Orthogonal Matrices In Exercises 25-32, determine whether the matrix is symmetric, orthogonal, both or neither. A=[001010100]28CR29CRDetermine Symmetric and Orthogonal Matrices In Exercises 25-32, determine wheter the matrix is symmetric, orthogonal, both, or neither. A=[4503501035045]31CR32CR33CR34CR35CR36CROrthogonally Diagonalizable Matrices In Exercises 37-40, determine whether the matrix is orthogonally diagonalizable. [3112]38CROrthogonally Diagonalizable Matrices In Exercises 37-40, determine whether the matrix is orthogonally diagonalizable. [412010215]40CR41CR42CR43CR44CR45CROrthogonal Diagonalization In Exercises 41-46, find a matrix P that orthogonally diagonalizes A. Verify that PTAP gives the correct diagonal form. A=[120210005]47CR48CR49CR50CR51CR52CRSteady State Probability Vector In Exercises 47-54, find the steady state probability vector for the matrix. An eigenvector v of an nn matrix A is a steady state probability vector when Av=v and the components of v sum to 1. A=[0.70.10.10.20.70.10.10.20.8]54CR55CR56CR57CR58CR59CR60CR61CR62CR63CRa Find a symmetric matrix B such that B2=A for A=[2112] b Generalize the result of part a by proving that if A is an nn symmetric matrix with positive eigenvalues, then there exists a symmetric matrix B such that B2=A.Determine all nn symmetric matrices that have 0 as their only eigenvalue.66CR67CR68CR69CRTrue or False? In Exercises 69 and 70, determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. a An eigenvalue of a matrix A is a scalar such that det(IA)=0. b An eigenvector may be the zero vector 0. c A matrix A is orthogonally diagonalizable when there exists an orthogonal matrix P such that P1AP=D is diagonal.71CR72CR73CR74CR75CR76CR77CR78CR79CR80CR81CR82CR83CR84CR85CR86CR87CR88CR1CMIn Exercises 1 and 2, determine whether the function is a linear transformation. T:M2,2R, T(A)=|A+AT|Let T:RnRm be the linear transformation defined by T(v)=Av, where A=[30100302]. Find the dimensions of Rn and Rm.4CMFind the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).Let T:R4R2 be the linear transformation defined by T(v)=Av, where A=[10100101]. Find a basis for a the kernel of T and b the range of T. c Determine the rank and nullity of T.In Exercises 7-10, find the standard matrix for the linear transformation T. T(x,y)=(3x+2y,2yx)8CM9CM10CM11CM12CM13CM14CM15CM16CM17CM18CMIn Exercises 19-22, find the eigenvalues and the corresponding eigenvectors of the matrix. [7223]20CM21CM22CMIn Exercises 23 and 24, find a nonsingular matrix P such that P-1AP is diagonal. A=[231012003]In Exercises 23 and 24, find a nonsingular matrix P such that P-1AP is diagonal. A=[0354410004]Find a basis B for R3 such that the matrix for the linear transformation T:R3R3, T(x,y,z)=(2x2z,2y2z,3x3z), relative to B is diagonal.Find an orthogonal matrix P such that PTAP diagonalizes the symmetric matrix A=[1331].Use the Gram-Schmidt orthonormalization process to find an orthogonal matrix P such that PTAP diagonalizes the symmetric matrix A=[022202220].28CM29CM30CM31CMProve that if A is similar to B and A is diagonalizable, then B is diagonalizable.Using Mathematical Induction In Exercises 1-4, use mathematical induction to prove the formula for every positive integer n. 1+2+3+...+n=n(n+1)22E3E4E5E6E7E8E9E10E11E12E13E14EUsing Proof by Contradiction In Exercises 1526, use proof by contradiction to prove the statement. If p is an integer and p2 is odd, then p is odd. Hint: An odd number can be written as 2n+1, where n is an integer.16E17E18E19E20E21E22E23E24E25E26E27E28E29E30E31E32E33E