Matrices Exercises And Solutions Pdf i Do the 8 column vectors in the matrix Hform a basis in R8 Prove or disprove ii Calculate HH T where denotes transpose Compare the results from i and ii and discuss Problem 3 Show that any 2 2 complex matrix has a unique represen tation of
Math 10B Spring 2018 Matrix Algebra Worksheet 1 Solutions i Write 2 2 matrices A B C such that AB AC but B 6 C Solution First a helpful observation Suppose A is invertible and AB AC Then multiplying both sides by A 1 gives A 1 AB A 1 AC Using distributivity and the fact that A 1A I this implies that IB IC and therefore B C 1 Math 262 Exercises and Solutions 1 Let A be a 3 215 3 matrix with real entries Prove that if A is not similar over R to a triangular matrix then A is similar over C to a diagonal matrix Proof Since A is a 3 215 3 matrix with real entries the characteristic polynomial f x of A is a polynomial of degree 3 with real coe cients We know
Matrices Exercises And Solutions Pdf
Matrices Exercises And Solutions Pdf
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A conjugate transpose of matrix A Ay conjugate transpose of matrix A notation used in physics A 1 inverse of square matrix A if it exists I n n nunit matrix I unit operator 0 n n nzero matrix AB matrix product of m nmatrix A and n pmatrix B A B Hadamard product entry wise product of m nmatrices Aand B A B AB BA commutator for square Problem Sets with Solutions MIT OpenCourseWare is a web based publication of virtually all MIT course content OCW is open and available to the world and is a permanent MIT activity
EXERCISES OF MATRICES OPERATIONS SOLUTIONS Throughout we assume that the dimensions of the matrices in this note make sense Question 1 Which of the following statements must be true 1 F Counterexample A 0 1 0 1 2 F 3 F Counterexample A 0 1 1 0 4 F Counterexample A 1 0 0 1 5 F Counterexample A 0 1 0 0 Use it in a variation of parameters solution of the equation u Au et 0 with initial condition u 0 1 0 Begin by nding some particular solution without regard to initial condition If you are careful and keep your matrices factorized you can use matrices
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The material taught in this book requires an active engagement with the exercises and we urge you not to read the solutions in advance Rather you should use the ones in EXERCISES OF MATRICES OPERATIONS Throughout we assume that the dimensions of the matrices in this note make sense Question 1 Which of the following statements must be true 1 If A2 A then Amust be either the identity matrix or the zero matrix 2 If Ais a 2 215 2 matrix and A 4 then 2A 8 3 If AT A then A 0 4 If A2
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