Dr. Byrne Math 237Fall 2008 Section 1Linear Algebra Practice Exam #3Exam format will be Long Answer Questions only. Show all work to receive full credit. While a calculator is not necessary for this exam you are encouraged to use a calculator to verify your steps. This practice exam is approximately twice as long as the exam will be. 1. Find the inverse of A using A-1 = )()det(1AadjA.A=0111011102. For the matrix A,a) Show |A|=35 using expansion by cofactors.1000000070000002603010001000000050001000107654321A b) Show |A|=35 using row reduction. 1000000070000002603010001000000050001000107654321A3. Answer part (a) and (b) below for an n x n matrix A.a) Write the definition of an eigenvector of A.b) Find the eigenvalues of A = .3000208364. Answer part (a) and (b) below.a) The eigenvalues of A = 300020836 are 6, -2 and -3. Find the eigenvectors and eigenspaces of A.b) Without diagonalizing A, determine if A is diagonalizable. Explain your reasoning.5. If A is an n x n matrix, is an eigenvalue for A if and only if p()=det(A-I)=0. Explain the origin of the characteristic polynomial p() by deriving in detail that p()=det(A-I). 6. The eigenvectors of A=455235224 are 111,110, 011.a) Write down a matrix P that diagonalizes A.b) Given the P you wrote down in part (a) above, write down D, the diagonalization of A where D=P-1AP.7. Find the complex eigenvalues of 1423A.8.
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