MATH 260 LINEAR ALGEBRA EXAM II Fall 2013 Instructions: The use of built-in functions of your calculator, such as det( ) or RREF, is prohibited.
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1 MAH 60 LINEAR ALGEBRA EXAM II Fall 0 Instructions: he use of built-in functions of your calculator, such as det( ) or RREF, is prohibited ) For the matrix find: a) M and C b) M 4 and C 4 ) Evaluate the determinant of the matrix below by cofactor expansion along the first row ) Evaluate the determinant of the matrix by using elementary row operations ) Let A, B, C be 4 4 matrices with A, B 0, and C Find and simplify each of the following, or briefly explain why you can t find the requested item: det AC b) [ points] A B a) [ points] det c) [ points] det AC d) [ points] det AB AC e) [ points] det 5) Use the adjoint method to find the inverse of the matrix below Math 60 Exam Fall 0 Condensed/Page of 5
2 6) Using a specific determinant, assess whether the points,,,,,, and 4,0, are coplanar 0,6,5, 7) Each part of this question is worth two points here is no partial credit for any of the parts You do not need to show any work Circle the letter(s) corresponding to the correct answer(s) for each part If you believe more than one choice is correct, mark all which apply If you attempt at least one part of this question, all of the parts will be graded 7) Say that A, B are both matrices, with A and following are FALSE? Mark all which apply a) A B b) AB is nonsingular det AB 0 c) d) A det det e) AB B Which of the matrix with A 7) Say that A is an FALSE? Mark all which apply a) det A b) A 6 det 64 c) A d) AA det 4 Which of the following are det 08 det e) he system Ax b will have a unique solution for any column vector b (Continued on the next page) Math 60 Exam Fall 0 Condensed/Page of 5
3 ) Say that A 5 0 Which of the following are FALSE? Mark all which apply a) A is singular b) det 5 A c) For this matrix A, C d) For this matrix A, M 0 e) For this matrix, det det A A ) Say that A Which of the following are FALSE? Mark all which apply a) If B is the matrix made by swapping the first and third rows of A, then det B 6 b) If B is the matrix made by performing the row operations R R, R R on A, then det B 6 c) If B is the matrix made by performing the row operations 4 R, on A, then det B d) If B is the matrix made by first swapping the first and fourth rows of A, det B 6 and then swapping the second and third rows of A, then e) If B is the matrix made by putting A into row echelon form, then det B (Continued on the next page) Math 60 Exam Fall 0 Condensed/Page of 5
4 75) Say that W is a nonempty subset of a vector space V Which of the following are FALSE? Mark all which apply a) If W contains the zero vector of V, then W is a subspace of V b) If V, V are subspaces of V, and, then W is a subspace of V c) If V, V are subspaces of V, and, then W is a subspace of V d) If W is a subspace of V, then W contains the zero vector of V e) If W contains only the zero vector of V, then W is a subspace of V 8) Let V be R, and uv, V, with scalars k R Let addition be defined as uv u v,uv and scalar multiplication be defined as ku ku,ku a) [5 points] Find the zero vector for V b) [5 points] Find the additive inverse vector for V 9) Let V be R, and uvw,, V, with scalars km, R Let addition be defined as uv u v,uv and scalar multiplication be defined as ku ku ku Is V a vector space?, 0) Let P be the vector space of polynomials, and let W be the nonempty subset of P containing polynomials p x such that p0 Is W a subspace of P under the usual polynomial addition and scalar multiplication? a a ) Is W abc,, b c R, a subset of M, a subspace under the usual matrix addition and scalar multiplication? ) Let u,, v,, and, w a) [4 points] On the graph below, sketch u, v, and u v Label everything clearly b) [6 points] Express w as a linear combination of u and v, showing all work Math 60 Exam Fall 0 Condensed/Page 4 of 5
5 ) BONUS [0 points] Determine whether the following statements are true or false If the statement is true, justify why it is true If the statement is false, explain why, or provide a counter-example a) W xy, x y is a subspace of addition and scalar multiplication W xy, x y is a subspace of addition and scalar multiplication b) R under the usual vector R under the usual vector 4) BONUS [0 points] Find the determinant using a cofactor expansion along the first row x y y y y x y y y y x y y y y x 5) BONUS [0 Points] Find det A using Dodgson s Condensation Method if A ) BONUS [0 Points] If P 0 is a plane in R, with the equation x yz 0, is P 0 a subspace of R, under the usual vector addition and scalar multiplication of R? Justify your answer Math 60 Exam Fall 0 Condensed/Page 5 of 5
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