Question 7. Consider a linear system A x = b with 4 unknown. x = [x 1, x 2, x 3, x 4 ] T. The augmented

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1 Question. How many solutions does x 6 = 4 + i have Practice Problems 6 d) 5 Question. Which of the following is a cubed root of the complex number i. 6 e i arctan() e i(arctan() π) e i(arctan() π)/3 6 e i(arctan() π+π)/3 d) Question 3. 4 If A = and B =, what is the product AB? 5 [ 5 9 AB = AB = 6 9 [ AB = d) AB = Question 4. 3 If A = and v = 4, what is A v? 4 A v = A v = A v = d) A v = [ 5 Question 5. If A R m n then what is the dimension of A T A m m Does not exist n n d) m n Question 6. Which of the following matrices is in echelon form?

2 Question 7. Consider a linear system A x = b with 4 unknown variables x = [x, x, x 3, x 4 T. The augmented matrix M = [A b has the reduced matrix, what are the pivot variables (basic variables)? x, x 3 x, x 4, d), Question 8. Consider a linear system A x = b with 4 unknown variables x = [x, x, x 3, x 4 T. The augmented matrix M = [A b has the reduced matrix, what are the free variables? x, x 3 x, x 4, d), Question 9. Which of the following matrices is in row canonical form? d) Question. What is the inverse of A = 4 A = / A =.5 / /4 A = / d) A =? 4 / Question. What is the inverse of A = [ 3 4 /3 A = A = A = d) A = 5? [ 4 3 [ 3 4

3 Question. What is the row canonical form of A = Already in row canonical form? d) Question 3. Given A = solve A x = 4 b for b = [, T. x = x = no solution d) infinite solutions 8 Question 4. How many solutions does A x = b have? The augmented matrix is [A b = no solution infinite solution insufficient information d) one solution Question 5. Given [A b = 3 5 how many solutions does A x = b have? no solution infinite solution insufficient information d) one solution Question 6. Given [A b = 3 5 how many solutions does A x = b have? no solution infinite solution insufficient information d) one solution 3

4 Question 7. Which of the following sets of vectors are independent? { [,,, 4 { }, 5 d) [, 3 } Question 8. Vector v = [ 5, T is a linear combination of which set of vectors? { } { [ 5 8,,, {} d) None of the above } Question 9. Vector v = [, T is a unique linear combination of which set of vectors? { } {},, { }, d) None of the above Question. Vector v = [ 3, T is in the span of what set? { } {} 6, 5 { }, Question. Which matrix below has colsp(a) R 3?

5 Question. Which matrix below has colsp(a) = R 3? d) None of the above Question 3. What is the rank of matrix A = ? 3 d) 6 Question 4. Which of the following sets is a basis for rowsp(a), where A = equivalent to 3 8 T,? T, T T T T 3, 4, 5 T, T,, which is row T Question 5. Which of the following sets is a basis for colsp(a), where A = {[ 3 [, } [ 5 { },, d) None of the above 5 T, 5 T?

6 Question 6. What is dim(rowsp(a)), where A = 3 5 5? 3 d) 4 Question 7. What is dim(colsp(a)), where A = 3 5 5? 3 d) 4 Question 8. What is the dim(ker(a)) if A = d) 3 e) 4? Question 9. Which of the following vectors is in the null space of A = Question 3. Which of the following matrices has det(a) = 6? 3 A = 4 A = 6 A = d) A = [ 4? 6 d) 6

7 Question 3. Which of the following matrices has det(a) =? A = A = A = Question 3. Compute the determinant of A = d)? e) None of the above Question 33. Let A and B be 4 4 matrices with det(a) = 3 and det(b) =, compute det(ba T ) / d) -/3 e) Need more information Question 34. Which of the following matrices has eigenvalues λ =, λ = 6, λ 3 =? 4 A = A = A = 8 d) A =

8 Question 35. Which of the following matrices has at least one eigenvalue λ =? 4 A = A = A = d) A = Question 36. A matrix A R 3 3 has eigenvalues λ =, λ =, λ 3 = 3. Which of the following statements must be true? d) Matrix A has 3 linearly independent eigenvectors Matrix A is full rank The reduced row canonical form of A has three pivot points All of the above Question 37. Which of the following vectors (if any) are eigenvectors of 4? Let u =, v = both u v d) neither e) Not enough information Question 38. How many distinct eigenvalues can a 3 3 matrix have? 3 d) 4 e) Question 39. Which of the following vectors are an orthogonal to v =? 8

9 / Question 4. Which of the following sets of vectors are an orthogonal basis for R 3? 3,,,, 4,,, Question 4. Which of the following matrices are invertible? 4 A = A = 6 6 A = Question 4. A matrix A R 3 3 is invertible. Which of the following statements must be true? Ax = b has a unique solution Matrix A has det The reduced row canonical form of A has three pivot points Question 43. If u =, v = 3 then u v = - d) 9

10 Question 44. If u = 3, v = then what is dist(u, v) d) 4 Question 45. If u = then u = d) Question 46. If there exists a matrix P such that D = P AP = 3 3, what is the dim(ker(a))? Not enough information Question 47. If there exists a matrix P such that A = P DP where D = of A? λ =, λ = 4, λ 3 = λ =, λ = 4, λ 3 =. λ =, λ =, λ 3 = i d) Not enough information what are the eigenvalues Question 48. What are the eigenvalues of A = [? λ =, λ = λ =, λ = 4. λ =, λ = d) Not enough information

11 Question 49. What is a possible transition matrix P that diagonalizes A = [?.5.5 A = A = A = d) A is not diagonalizable Question 5. If λ = 3.89 is an eigenvalue of A = d) 7 what is the det(3.89i A)?