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Unit 1: Matrix methods and linear systems - Practice Quiz

MTH165 — Mathematics For Engineers 60 Questions
0 Correct 0 Wrong 60 Left
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1 How many rows and columns does a matrix of order have?

review of matrices Easy
A. 3 rows and 2 columns
B. 2 rows and 2 columns
C. 2 rows and 3 columns
D. 3 rows and 3 columns

2 Which matrix has all diagonal entries equal to and all other entries equal to ?

review of matrices Easy
A. Scalar matrix
B. Zero matrix
C. Triangular matrix
D. Identity matrix

3 Which notation represents the interchange of the first and second rows?

elementary operations of matrices Easy
A.
B.
C.
D.

4 Which of the following is a valid elementary row operation?

elementary operations of matrices Easy
A. Multiplying a row by
B. Adding a new row
C. Multiplying a row by
D. Deleting a row

5 The rank of a matrix is equal to the maximum number of what kind of rows or columns?

rank of a matrix Easy
A. Nonzero rows or columns
B. Linearly independent rows or columns
C. Linearly dependent rows or columns
D. Equal rows or columns

6 What is the rank of the identity matrix ?

rank of a matrix Easy
A.
B.
C.
D.

7 A set of vectors is linearly dependent if one vector can be expressed as what?

linear dependence and independence of vectors Easy
A. A linear combination of the others
B. A scalar quantity
C. A product of the others
D. A zero determinant

8 Which pair of vectors in is linearly independent?

linear dependence and independence of vectors Easy
A. and
B. and
C. and
D. and

9 A linear system is consistent when it has what?

solution of linear system of equations Easy
A. No possible solution
B. Exactly two solutions
C. At least one solution
D. Only the zero solution

10 What is the solution of the system and ?

solution of linear system of equations Easy
A.
B.
C.
D.

11 If is an invertible square matrix, how many solutions does have for every ?

solution of linear system of equations Easy
A. Exactly two solutions
B. No solutions
C. Exactly one solution
D. Infinitely many solutions

12 For an invertible matrix , which equation defines its inverse ?

inverse of matrices Easy
A.
B.
C.
D.

13 What is the inverse of the diagonal matrix ?

inverse of matrices Easy
A.
B.
C.
D.

14 If satisfies , what is called?

eigenvalues and eigenvectors Easy
A. An eigenvalue
B. A matrix rank
C. A row vector
D. A determinant

15 For , which vector is an eigenvector corresponding to the eigenvalue ?

eigenvalues and eigenvectors Easy
A.
B.
C.
D.

16 The sum of the eigenvalues of a square matrix, counting multiplicities, equals which quantity?

properties of eigenvalues Easy
A. The trace of the matrix
B. The order of the matrix
C. The rank of the matrix
D. The norm of the matrix

17 The product of the eigenvalues of a square matrix, counting multiplicities, equals which quantity?

properties of eigenvalues Easy
A. Its rank
B. Its determinant
C. Its trace
D. Its dimension

18 If is an eigenvalue of , what is the corresponding eigenvalue of ?

properties of eigenvalues Easy
A.
B.
C.
D.

19 What does the Cayley-Hamilton theorem state about a square matrix ?

Cayley-Hamilton theorem Easy
A. satisfies its characteristic equation
B. always has distinct eigenvalues
C. always has an inverse
D. equals its transpose

20 If the characteristic polynomial of is , which equation follows from the Cayley-Hamilton theorem?

Cayley-Hamilton theorem Easy
A.
B.
C.
D.

21 If and , what is ?

review of matrices Medium
A.
B.
C.
D.

22 For , the operation produces a matrix . What is ?

elementary operations of matrices Medium
A.
B.
C.
D.

23 What is the rank of ?

rank of a matrix Medium
A.
B.
C.
D.

24 For what value of does have rank ?

rank of a matrix Medium
A.
B.
C.
D.

25 Which set of vectors is linearly dependent?

linear dependence and independence of vectors Medium
A.
B.
C.
D.

26 For which values of are the vectors , , and linearly independent?

linear dependence and independence of vectors Medium
A.
B. only
C. only
D.

27 Solve the system and .

solution of linear system of equations Medium
A.
B.
C.
D.

28 For what value of does the system and have infinitely many solutions?

solution of linear system of equations Medium
A.
B.
C.
D.

29 Find the solution of , , and .

solution of linear system of equations Medium
A.
B.
C.
D.

30 A homogeneous system has unknowns and a coefficient matrix of rank . Which statement describes its solution set?

solution of linear system of equations Medium
A. It has one free parameter.
B. It has two free parameters.
C. It has a unique nonzero solution.
D. It has no solution.

31 What is the inverse of ?

inverse of matrices Medium
A.
B.
C.
D.

32 If and , what is ?

inverse of matrices Medium
A.
B.
C.
D.

33 What are the eigenvalues of ?

eigenvalues and eigenvectors Medium
A.
B.
C.
D.

34 Which vector spans the eigenspace of corresponding to ?

eigenvalues and eigenvectors Medium
A.
B.
C.
D.

35 What are the eigenvalues of ?

eigenvalues and eigenvectors Medium
A. and
B. and
C. and
D. and

36 A matrix has trace and determinant . If two eigenvalues are and , what is the third eigenvalue?

properties of eigenvalues Medium
A.
B.
C.
D.

37 If the eigenvalues of an invertible matrix are and , what are the eigenvalues of ?

properties of eigenvalues Medium
A. and
B. and
C. and
D. and

38 If is an eigenvalue of , what is the corresponding eigenvalue of ?

properties of eigenvalues Medium
A.
B.
C.
D.

39 Which matrix equation follows from the Cayley-Hamilton theorem for ?

Cayley-Hamilton theorem Medium
A.
B.
C.
D.

40 An invertible matrix satisfies . Which expression equals ?

Cayley-Hamilton theorem Medium
A.
B.
C.
D.

41 Let be matrices, with invertible. If and what is the determinant of the block matrix ?

review of matrices Hard
A.
B.
C.
D.

42 A matrix has . Matrix is obtained by successively applying , , , and . What is ?

elementary operations of matrices Hard
A.
B.
C.
D.

43 For which statement correctly describes its rank?

rank of a matrix Hard
A. for and otherwise
B. for and otherwise
C. only for and otherwise
D. only for and otherwise

44 Let , , and . Which statement is correct?

linear dependence and independence of vectors Hard
A. The vectors are independent exactly when
B. The vectors are independent exactly when
C. The vectors are dependent for every real
D. The vectors are dependent exactly when

45 Consider Which classification is correct?

solution of linear system of equations Hard
A. Unique for every except ; infinitely many when and
B. Unique if ; infinitely many for or ; none otherwise
C. Unique if ; infinitely many for or ; none otherwise
D. Unique for every except ; infinitely many when and

46 Let and . What is the inverse of ?

inverse of matrices Hard
A.
B.
C.
D.

47 For which statement is correct?

eigenvalues and eigenvectors Hard
A. The eigenvalue has algebraic multiplicity and a two-dimensional eigenspace
B. The eigenvalue has algebraic multiplicity and eigenspace
C. The matrix has eigenvalues , , and with distinct eigenvectors
D. The eigenvalue has algebraic multiplicity and eigenspace

48 Let be a real matrix of rank , and let be a matrix. If is consistent, what is the dimension of its affine solution set in the space of matrices?

solution of linear system of equations Hard
A.
B.
C.
D.

49 Elementary matrices represent row operations applied to an invertible matrix in that order. If the operations reduce to , which expression equals ?

elementary operations of matrices Hard
A.
B.
C.
D.

50 In , let , , , and . For which values of are the four vectors linearly independent?

linear dependence and independence of vectors Hard
A. Exactly when
B. Exactly when
C. Exactly when
D. Exactly when

51 Let and . If what is the dimension of the set of vectors satisfying both and ?

rank of a matrix Hard
A.
B.
C.
D.

52 Let be invertible and let be nonzero vectors satisfying . Which vector is guaranteed to be a nonzero null vector of ?

inverse of matrices Hard
A.
B.
C.
D.

53 For what is the inertia of when ? The inertia lists the numbers of positive, negative, and zero eigenvalues.

eigenvalues and eigenvectors Hard
A.
B.
C.
D.

54 An invertible matrix has eigenvalues , , and , counted with algebraic multiplicity. What is ?

properties of eigenvalues Hard
A.
B.
C.
D.

55 Let satisfy , and suppose has distinct eigenvalues. Which statement must hold?

properties of eigenvalues Hard
A. Every eigenvector of is also an eigenvector of
B. The eigenvalues of and have equal multiplicities
C. Every eigenvector of is also an eigenvector of
D. The characteristic polynomials of and are equal

56 A matrix has characteristic polynomial . Which expression equals ?

Cayley-Hamilton theorem Hard
A.
B.
C.
D.

57 An invertible matrix has characteristic polynomial . Which formula for follows from the Cayley–Hamilton theorem?

Cayley-Hamilton theorem Hard
A.
B.
C.
D.

58 Let be a matrix of rank and a matrix of rank . What is the smallest value that can attain?

rank of a matrix Hard
A.
B.
C.
D.

59 Let have rank , and suppose with . Which statement correctly classifies ?

solution of linear system of equations Hard
A. It has no solution if , and a unique solution otherwise
B. It has infinitely many solutions for every vector regardless of
C. It has a unique solution if , and infinitely many solutions otherwise
D. It has infinitely many solutions with one free parameter if , and no solution otherwise

60 A matrix has characteristic polynomial . If and , which conclusion is correct?

eigenvalues and eigenvectors Hard
A. is not diagonalizable because the eigenspace for has dimension
B. is diagonalizable because both eigenvalues have algebraic multiplicity
C. is not diagonalizable because the eigenspace for has dimension
D. is diagonalizable because the two stated ranks are different
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Unit 2