Sample Quiz on Eigenspaces Question # 1: The eigenvalues of the matrix A:=matrix([[1,1,2],[0,-1,3],[0,0,4]]); Select Answer Here (a) are 1,1,2 (b) are 1,-1,4 (c) are 1,3,4 (d) are 1,3,-4 Question # 2: The eigenvalues of an invertible matrix A are:1,-2,3 What are the eigenvalues of the inverse of the matrix A? Select Answer Here (a) -1,2,-3 (b) 1,1/2,1/3 (c) -1,1/2,-1/3 (d) 1,-1/2, 1/3 Question # 3: The eigenvalues of a matrix A are: 1,2,-2. Then the eigenvalues of the fifth power of matrix A are: Select Answer Here (a) 1,32,-32 (b) 1,-2,2 (c)-1,16,-16 (d) 1,-10, 10 Question # 4: The eigenvalues of a symmetric matrix with real entries Select Answer Here (a) are all complex (b) are all real (c) some are real and some are complex (d) none of the above Question # 5: If the geometric and the algebraic multiplicities of the eigenvalues of a matrix are equal, then Select Answer Here (a) the matrix is diagonalizable (b) the matrix is not diagonalizable (c) we cannot tell if the matrix is diagonalizable (d) non of the above Question # 6: The vector w:= (A^3)*v where v and A are given by: v:=vector([10,10]); A:=matrix([[4,0],[2,2]]): Select Answer Here (a) is [40,40] (b) is [640,640] (c) is [320,320] (d) is [10,10] Question # 7: If the dimension of the solution space of a homogeneous system of 3 equations in 3 unknowns is 2, then the eigenvalue 0 of the coefficient matrix has: Select Answer Here (a) algebraic multiplicity 2 and geometric multiplicity 1 (b) algebraic multiplicity 2 and geometric multiplicity 2 (c) algebraic multiplicity 1 and geometric multiplicity 1 (d) algebraic multiplicity 1 and geometric multiplicity 2 Question # 8: The characteristic polynomial of a given matrix is p(x)=x^3-3*x^2+2*x. Then the matrix Select Answer Here (a) is diagonalizable (b) is not diagonalizable (c) must be a symmetric matrix (d) must be a diagonal matrix whose diagonal entries 0,1,2