a 3 marks Is Ker A Explain your answer 4b 3 marks Is A invertible Why 4c 3

# Linear Algebra with Applications (3rd Edition)

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4a.[3 marks]Is Ker(A) ={0}? Explain your answer. 4b.[3 marks]IsAinvertible? Why? 4c.[3 marks]DoesAhave linearly independent eigenvectors? Explain. 4d. [6 marks] Give a specific example of a matrix A satisfying the above three properties and whose eigenvalues add up to zero. Sol. [4a.] A has linearly dependent columns: for example, it follows from conditions i. and ii. that A ( 2 0 0 + 0 - 1 0 ) = 0 . This implies that Ker( A ) 6 = { 0 } . [4b.] A is not invertible as Ker( A ) 6 = { 0 } . [4c.] Yes, the eigenvectors of a symmetric matrix are linearly independent (and can be chosen to be orthonormal). [4d.] For example A = 1 2 - 3 2 4 - 6 - 3 - 6 a 33 , with a 33 = - 5. In fact: A 2 0 0 gives 2 times the first column of A and A 0 - 1 0 gives - 1 times the second column of A . Then, from the symmetry condition iii. we get a 13 = a 31 and a 23 = a 32 . For a 33 we impose λ k = trace( A ) = 1 + 4 + a 33 = 0. From this relation we deduce a 33 = - 5.
5. [10 marks] Let A = 1 2 2 4 . 5a. [2 marks] Find the eigenvalues of A . 5b. [3 marks] Give a factorization A = QDQ T where Q has orthonormal columns and D is a diagonal matrix. 5c. [4 marks] As t → ∞ , what is the limit of u ( t ) for du ( t ) dt = - Au ( t ) given the initial condition u (0) = 3 1 ? 5d.[1 marks]IsAa positive definite matrix?Why?Give the quadratic form
6. [10 marks] 6a. [3 marks] If possible, find an invertible matrix M such that M - 1 1 1 1 1 1 1 1 1 1 M = 1 1 1 1 2 2 1 2 2 . If it is not possible, state why M cannot exist.