MATH 2360 REVIEW PROBLEMS
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1 MATH 2360 REVIEW PROBLEMS Problem 1: In (a) (d) below, either compute the matrix product or indicate why it does not exist: ( )( ) (a) ( ) (b) (c) 0 3 ) 1 4 ( ( ) 2 1 (d) Problem 2: Solve the linear system x 2 + x 3 = 14 2x 1 + x 2 +2x 3 = 14 3x 1 +4x 2 +2x 3 = 14 Problem 3: (a) Find the general solution for the linear system 2x 1 + x 2 x 3 = 2 x 1 2x 2 +3x 3 = 2 5x 1 + x 3 = 6 (b) Solve the linear system 2x 1 + x 2 x 3 = 1 x 1 2x 2 +3x 3 = 1 5x 1 + x 3 = 1 1
2 2 MATH 2360 REVIEW PROBLEMS (c) Is it possible for a linear system of the form 2x 1 + x 2 x 3 = b 1 x 1 2x 2 +3x 3 = b 2 5x 1 + x 3 = b 3 (with b 1,b 2,b 3 known) to have exactly one solution? Why? Problem 4: For each of the following matrices, determine whether it is a reduced row echelon matrix: ( ) (a) is row echelon, but not reduced row echelon (b) is reduced row echelon (c) is reduced row echelon (d) is not row echelon (e) is not row echelon Problem 5: Which of the following matrix product can be computed? ( Compute )( them. ) (a) ( ) (b) (c) ( ) (d) 1 2 ( ) 3
3 MATH 2360 REVIEW PROBLEMS 3 Problem 6: Solve the linear system x 1 + 2x 2 + 3x 3 = 4 5x 1 + 6x 2 + 7x 3 = 8 9x 1 +10x 2 +11x 3 = 12 Problem 7: (a) Write down the augmented matrix of the linear system x 1 +2x 2 +3x 3 = 4 2x 1 +3x 2 +4x 3 = 5 3x 1 +4x 2 +5x 3 = 6 (b) Verify that (x 1,x 2,x 3 ) = (0, 1,2) is a solution to the system in (a). (c) The reduced row echelon form of the coefficient matrix of the system in (a) is (You are not required to verify this!) How many solutions does the system have? (And why?) Problem 8: In (a) (d) below, either compute the matrix product or indicate ( why it )( does not exist: ) (a) (b) ) ( ( ) (c) ( )( ) (d)
4 4 MATH 2360 REVIEW PROBLEMS Problem 9: (a) Write down the augmented matrix of the linear system x 1 + 2x 2 x 3 = 0 2x 1 + 2x 2 + 2x 3 = 0 2x 2 4x 3 = 0 (b) Verify that (x 1,x 2,x 3 ) = ( 3,2,1) is a solution to the system in (a). (c) How many solutions does the system in (a) have? Answer and explain without row reducing or solving the system. Problem 10: In (a) (d) below, either compute the matrix product or indicate ( why )( it does not ) exist: (a) ( )( ) (b) ( ) (c) ( ) 1 1 (d) Problem 11: In (a) (d) below, mark each statement true or false. If it is false, give an counterexample. (a) If A 2 = I, then A is invertible. (b) If the reduced row echelon form of A has a pivot in every column, then the linear system Ax = b is consistent. (c) If the number of equations in a linear system is greater than the number of variables, then the system cannot have more than one solution. (d) A square matrix A is invertible, if and only if A can be row reduced to the identity matrix I. Problem 12: Consider the linear system x 1 + x 3 = 2 2x 1 +x 2 4x 3 = 3 4x 1 +x 2 3x 3 = 1
5 MATH 2360 REVIEW PROBLEMS 5 (a) Let A be the coefficient matrix of this system. Find A, show that it is invertible, and compute A 1. (b) Solve the linear system. Problem 13: (a) Find the general solution for the linear system (b) Solve the linear system 2x 1 + x 2 x 3 = 2 x 1 2x 2 +3x 3 = 2 5x 1 + x 3 = 6 2x 1 + x 2 x 3 = 1 x 1 2x 2 +3x 3 = 1 5x 1 + x 3 = 1 (c) Is it possible for a linear system of the form 2x 1 + x 2 x 3 = b 1 x 1 2x 2 +3x 3 = b 2 5x 1 + x 3 = b 3 (with b 1,b 2,b 3 known) to have exactly one solution? Why? Problem 14: (a) Find the general solution for the linear system (b) Solve the linear system x 1 + x 2 x 3 = 1 2x 1 + x 2 3x 3 = 2 3x 2 +3x 3 = 0 x 1 + x 2 x 3 = 0 2x 1 + x 2 3x 3 = 1 3x 2 +3x 3 = 1 (c) Is it possible for a linear system of the form x 1 + x 2 x 3 = b 1 2x 1 + x 2 3x 3 = b 2 3x 2 +3x 3 = b 3 (with b 1,b 2,b 3 known) to have exactly one solution? Why/why not?
6 6 MATH 2360 REVIEW PROBLEMS Problem 15: The matrix A = has the reduced row echelon form R = (a) Determine whether or not the linear system with augmented matrix A is consistent. If it is consistent, find the solution(s). (b) Determine whether or not the homogeneous linear system with coefficient matrix A is consistent. If it is consistent, find the solution(s). (c) Use the information given above to determine whether the linear system is consistent or not. x 1 +2x 2 =7 x 1 x 2 +x 3 x 4 = 4 2x 1 +5x 2 +x 3 x 4 =17 Problem 16: (a) Write down the augmented matrix of the linear system x 1 + x 3 + 4x 4 = 1 2x 1 + x 2 x 3 = 2 x 1 + x 2 + 2x 3 + 2x 4 = 1 (b) Verify that (x 1,x 2,x 3,x 4 ) = (0,1, 1,0) is a solution to the system in (a). (c) How many solutions does the system in (a) have? Answer and explain without row reducing or solving the system. Problem 17: In (a) (d) below, either compute the matrix product or indicate ( why )( it does ) not exist: (a) ( )( ) (b)
7 ( )( ) (c) ( ) (d) MATH 2360 REVIEW PROBLEMS 7 Problem 18: (a) Show that a linear system with coefficient matrix always has exactly one solution. A = (b) Is it possible for a linear system with coefficient matrix to be inconsistent? Problem 19: In (a) (d) below, mark each statement true or false. If it is false, give an counterexample. (a) If A is an upper triangular matrix, then A is invertible. (b) If the reduced row echelon form of A has a zero row, then the linear system Ax = b is inconsistent. (c) A diagonal matrix is symmetric. (d) An underdetermined linear system cannot have exactly one solution. Problem 20: In (a) (d) below, find an elementary matrix E such that B = EA. (a) A = , B = ( ) ( ) (b) A =, B = (c) A = , B =
8 8 MATH 2360 REVIEW PROBLEMS (d) A = , B = Problem 21: For each of the matrices below, determine whether it is singular or invertible. Make sure to explain your reasoning. If it invertible, find the inverse. ( ) 1 3, , ( ) 1 2, 2 1 ( ) 1 1, Problem 22: In (a) (c) below, mark each statement true or false. If it is false, give an example showing why. (a) If A is a square matrix, then A has a multiplicative inverse. (b) The product EF of two elementary n n matrices E and F is again an elementary matrix. (c) If A and B are invertible n n matrices, then the product AB is again invertible. Problem 23: Let A be an n n invertible matrix. Mark the following statements true or false: (a) A 2 is invertible. (b) If x 0, then Ax 0. (c) The homogeneous system Ax = 0 has no solutions. ( ) 2 3 Problem 24: (a) Is the matrix invertible? If so, find the 3 5 inverse. (b) Is the matrix invertible? If so, find the inverse Problem 25: Let A = (a) Compute A 2. (b) Show that A is invertible, and find A 1.
9 Problem 26: Let (a) Compute A 1. (b) Solve the linear system Problem 27: Let (a) Compute A 1. (b) Solve the linear system Problem 28: Let (a) Compute A 1. (b) Solve the linear system MATH 2360 REVIEW PROBLEMS 9 A = x 1 +3x 2 x 3 = 1 x 1 2x 2 = 2 2x 1 +6x 2 x 3 = A = x 1 +2x 2 +3x 4 = 1 x 2 +2x 3 +3x 4 = 2 x 1 +2x 4 = 0 3x 1 +3x 2 + x 4 = 2 A = x 1 +2x 2 +3x 3 = 2 x 1 2x 3 = 1 x 1 x 2 + x 3 = 1 Problem 29: Let A =
10 10 MATH 2360 REVIEW PROBLEMS (a) Compute A 1. (b) Solve the linear system 2x 1 +2x 2 x 3 = 1 4x 1 + x 2 2x 3 = 1 3x 1 + x 3 = 1 Problem 30: Let A = (a) Compute A 1. (b) Solve the linear system 2x 1 +2x 2 +x 3 = 1 x 1 +x 3 = 1 x 1 + x 2 +x 3 = 1 Problem 31: Let A = (a) Compute A 3. (b) Show that A is invertible, and find A 1. Problem 32: Let A be a 3 3 matrix. Is the set {x R 3 Ax = 2x} a subspace of R 3? (Justify your answer.) Problem 33: The 3 5 matrix A has reduced row echelon form R =
11 MATH 2360 REVIEW PROBLEMS 11 (a) The first and third columns of A are a 1 = 2 2 and a 3 = Find A. (b) Find a basis for the nullspace of A. (c) Find a basis for the row space of A. Problem 34: (a) Define the rank and the nullity of a matrix A. (b) Let A be a 5 6 matrix with rank 3. What is the dimension of N(A)? (c) Let A be a 4 6 matrix, and assume that the reduced row echelon form of A has one zero row. What is the dimension of the column space for A? (d) Assume that the vectors a 1,a 2,a 3,a 4 R 4 are linearly independent. Will they span R 4? (Justify your answer.) Problem 35: Let A = and R = Then R is the reduced row echelon form of A. (You do not need to check this.) (a) Find a basis for the column space of A. (b) Find a basis for the row space of A. (c) Find a basis for the nullspace of A.
12 12 MATH 2360 REVIEW PROBLEMS Problem 36: The matrices A and R are given by A = R = Also, R is the reduced row echelon form of A. (You do not need to verify this.) (a) Find a basis for the column space of A. (b) Find a basis for the row space of A. (c) Find a basis for the nullspace of A. Problem 37: Let the matrices A and R be given by A = , R = Then R is the reduced row echelon form of A. (You do not need to check this.) (a) Find a basis for the nullspace of A. (b) Find a basis for the column space of A. (c) Find a basis for the row space of A. (d) Solve the linear system 2x 1 + 4x 2 + x 3 = 0 x 1 + x 2 + x 3 = 3 x 2 +2x 3 = 6 2x 1 + x 2 +4x 3 = 12 directly, without performing any row reductions. (It is possible using the information above.) Problem 38: Let A and B be 3 3 matrices. Determine whether is a subspace of R 3. U = {x R 3 Ax = Bx} Problem 39: In (a) (c) below, mark each statement true or false. If it is false, give a counterexample.
13 MATH 2360 REVIEW PROBLEMS 13 (a) If v 1,...,v 4 R 4 are linearly independent, then they form a basis for R 4. (b) If A is an m n matrix, then the nullspace N(A) is a subspace of R m. (c) The column space and the row space for a matrix A have the same dimension. Problem 40: The reduced row echelon form of the matrix A = is R = (a) Find a basis for the nullspace N(A). (b) Find a basis for the row space of A. (c) Find a basis for the column space of A. Problem 41: Let A be an m n matrix, and let b R m. Mark the following statements true or false: (a) If the system Ax = b is consistent, then the columns of A span R m. (b) IfthesystemAx = bhasexactlyonesolution,thenthecolumns of A are linearly independent. (c) If the columns of A are linearly independent, then the system Ax = b has exactly one solution. (d) If the columns of A span R m, then the system Ax = b is consistent. Problem 42: Let the matrices A and R be given by A = , R =
14 14 MATH 2360 REVIEW PROBLEMS Then R is the reduced row echelon form of A. (You do not need to check this.) (a) Find a basis for the nullspace of A. (b) Find a basis for the column space of A. (c) Find a basis for the row space of A. Problem 43: (a) Let A be an m n matrix. Define the nullspace for A: { } N(A) = Let A = The reduced row echelon form of A is then R = (You do not need to verify this!) (b) Write down the solution to Ax = 0 in terms of free variables. (c) Use the result from (b) to find a basis for N(A). (d) What is the dimension of N(A)? Let x 1 = 2 6, x 2 = 3 7, x 3 = 4 8, x 4 = (e) Are the vectors x 1,x 2,x 3,x 4 linearly independent? (Justify your answer.) (f) Do the vectors x 1,x 2,x 3,x 4 span R 3? (Justify your answer.) Problem 44: Let a R 3 be a particular vector. Prove that is a subspace of R 3 3. S = {X R 3 3 Xa = 0} Problem 45: In (a) (d) below, mark the statements as true or false. (a) A subspace contains the zero vector 0.
15 MATH 2360 REVIEW PROBLEMS 15 (b) If the vector space V is spanned by n vectors, then it has dimension n. (c) Three linearly independent vectors in R 3 will be a basis for R 3. (d) If A is a matrix, then the dimension of N(A) is equal to the number of columns in A with a pivot. Problem 46: Let u = 1 1, v = Show that B = {u,v,w} is a basis for R 3., w = Problem 47: (a) Let A be an m n matrix. Define the rank of A. (b) Let A is an 8 7 matrix of rank 4. What is the dimension of the nullspace N(A)? (c) Let A and B be two row-equivalent m n matrices. Do A and B have the same nullspace? The same column space? The same row space? Problem 48: In (a) and (b) below, mark the statements as true or false. (a) For any matrix A, the row and column spaces have the same dimension. (b) If A is an m n matrix, then the rank of A plus the nullity of A equals the number of rows in A. Problem 49: In (a) (d) below, determine whether S is a subspace of V. Justify your answer. (a) S = {(x 1,x 2 ) T R 2 2x 1 +3x 2 = 1}, V = R 2. (b) S = {A R 2 2 a 11 +a 12 +a 21 +a 22 = 0}, V = R 2 2. (c) S = {p(x) P 4 p (0) = 0}, V = P 4. (d) S = {(x 1,x 2,x 3 ) R 3 x 1 x 2 = 0}, V = R 3. Problem 50: In (a) (d) below, determine whether S is a subspace of V. Justify your answer.
16 16 MATH 2360 REVIEW PROBLEMS (a) S = {A R 2 2 det(a) = 0}, V = R 2 2. (b) S = {x R 3 x 1}, V = R 3. (c) S = {x R 3 x 1 +x 2 = 0}, V = R 2. (d) S = {f(x) P 4 f(1) = 1}, V = P 4. Problem 51: Suppose that A is a 4 7 matrix of rank 4. In (a) (c) below, determine whether the statement is true or false. (a) The rows of A span R 1 7. (b) N(A) has dimension 3. (c) The columns of A span R 4.
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