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Unit 1: Matrix methods and linear systems - Practice Quiz

MTH165 — Mathematics For Engineers 60 Questions
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1 What is the order of the matrix ?

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

2 Which matrix is called a square matrix?

Review of matrices Easy
A. A matrix with equal columns
B. A matrix with zero entries
C. A matrix with equal rows
D. A matrix with equal rows and columns

3 What is the transpose of ?

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

4 Which operation is an elementary row operation?

Elementary operations of matrices Easy
A. Multiplying rows together
B. Squaring every entry
C. Adding two columns
D. Interchanging two rows

5 What happens when a row is multiplied by a nonzero scalar during an elementary row operation?

Elementary operations of matrices Easy
A. The matrix becomes square
B. The row is scaled
C. The row is deleted
D. The columns are reversed

6 Which row operation replaces by ?

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

7 The rank of a matrix is the maximum number of linearly independent __.

Rank of a matrix Easy
A. Rows or columns
B. Zero entries
C. Diagonal entries
D. Square submatrices

8 What is the rank of the zero matrix?

Rank of a matrix Easy
A. The number of rows
B. The number of columns
C.
D.

9 What is the rank of the identity matrix ?

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

10 Two vectors are linearly dependent if one vector is a __ of the other.

Linear dependence and independence of vectors Easy
A. Scalar multiple
B. Determinant
C. Inverse
D. Transpose

11 Which set of vectors is linearly dependent?

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

12 A set containing the zero vector is always:

Linear dependence and independence of vectors Easy
A. An orthogonal set
B. Linearly independent
C. Linearly dependent
D. A basis

13 A system of linear equations is consistent if it has:

Solution of linear system of equations Easy
A. Exactly two solutions
B. No solution
C. Only negative solutions
D. At least one solution

14 The augmented matrix of and is:

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

15 A homogeneous system of linear equations always has which solution?

Solution of linear system of equations Easy
A. Only a positive solution
B. Only a unique nonzero solution
C. No solution
D. The zero solution

16 For a square matrix , the inverse satisfies:

Inverse of matrices Easy
A. always
B.
C.
D.

17 A square matrix has an inverse only if its determinant is:

Inverse of matrices Easy
A. One only
B. Nonzero
C. Negative
D. Zero

18 For an eigenvalue and eigenvector , which equation holds?

Eigenvalues and eigenvectors Easy
A. for every
B.
C.
D.

19 The sum of the eigenvalues of a square matrix equals its:

Properties of eigenvalues Easy
A. Determinant
B. Trace
C. Rank
D. Inverse

20 The product of the eigenvalues of a square matrix equals its:

Properties of eigenvalues Easy
A. Rank
B. Trace
C. Determinant
D. Transpose

21 If and , what is ?

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

22 A matrix is obtained from a square matrix by interchanging two rows, multiplying one row by , and then adding twice one row to another. If , 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 are the vectors , , and linearly dependent?

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

25 Solve the system , , and .

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

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

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

27 What is the inverse of ?

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

28 If and are invertible matrices of the same order, which expression equals ?

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

29 What are the eigenvalues of ?

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

30 Which vector is an eigenvector of corresponding to the eigenvalue ?

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

31 An invertible matrix has eigenvalues , , and . What is ?

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

32 If the eigenvalues of are , , and , what are the eigenvalues of ?

Properties of eigenvalues Medium
A. , , and
B. , , and
C. , , and
D. , , and

33 Using the Cayley-Hamilton theorem, find for .

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

34 A matrix satisfies the characteristic equation . Which expression is equal to ?

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

35 For what value of does the matrix have rank ?

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

36 Given , , and , which relation is correct?

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

37 The reduced augmented matrix of a linear system is . Which is its general solution?

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

38 A matrix is obtained from a matrix by the column operations and . If , what is ?

Elementary operations of matrices Medium
A.
B. It cannot be determined
C.
D.

39 If is a matrix and is a matrix, which statement about is correct?

Review of matrices Medium
A. and has order
B. and has order
C. and has order
D. and has order

40 What are the eigenvalues, including algebraic multiplicities, of ?

Eigenvalues and eigenvectors Medium
A. twice and once
B. once and twice
C. twice and once
D. , , and once each

41 Let be a square matrix satisfying . Which expression equals ?

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

42 A matrix is obtained from a matrix by interchanging two rows, multiplying one column by , and adding five times one row to another. If , what is ?

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

43 Suppose is obtained from an matrix using elementary row operations. Which statement is always true?

Elementary operations of matrices Hard
A. and have the same row and column spaces, but their ranks need not be equal.
B. and have the same determinant and null space whenever .
C. and have the same null space and rank, but their column spaces need not be equal.
D. and have the same column space and rank, but their null spaces need not be equal.

44 For which values of does the matrix have rank exactly ?

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

45 Let be an matrix of rank . What is the rank of the block matrix

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

46 Consider , , , and . For which value of are these vectors linearly dependent?

Linear dependence and independence of vectors Hard
A.
B.
C.
D.

47 Let be a basis of a three-dimensional vector space , and let be linear with . Which statement about is correct?

Linear dependence and independence of vectors Hard
A. Their relation space is , and their span has dimension .
B. Their relation space is , and their span has dimension .
C. They are linearly independent because is a basis of .
D. Their relation space is , and their span has dimension .

48 For which pair does the system have infinitely many solutions?

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

49 For an matrix , suppose has a solution for every and has at most one solution for each . Which condition must hold?

Solution of linear system of equations Hard
A. and
B. and
C. and
D. and

50 Let and . What is the inverse of ?

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

51 Let , , and be invertible square matrices of the same order. Without assuming that and commute, which expression equals ?

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

52 Let , where and . Which description of the eigenvalues and eigenspaces of is correct?

Eigenvalues and eigenvectors Hard
A. The eigenvalues are ; and span the eigenspace.
B. The eigenvalues are ; the coordinate vectors form corresponding eigenvectors.
C. The eigenvalues are ; spans the eigenspace and is the zero eigenspace.
D. The eigenvalues are ; spans the eigenspace and is the zero eigenspace.

53 For which statement is correct?

Eigenvalues and eigenvectors Hard
A. The eigenspace for has dimension , the minimal polynomial is , and is not diagonalizable.
B. The eigenspace for has dimension , the minimal polynomial is , and is diagonalizable.
C. The eigenspace for has dimension , the minimal polynomial is , and is diagonalizable.
D. The eigenspace for has dimension , the minimal polynomial is , and is diagonalizable.

54 A real matrix has trace and one eigenvalue . What are its remaining eigenvalues and determinant?

Properties of eigenvalues Hard
A. The remaining eigenvalues are and , and the determinant is .
B. The remaining eigenvalues are and , and the determinant is .
C. The remaining eigenvalues are and , and the determinant is .
D. The remaining eigenvalues are and , and the determinant is .

55 Let be a matrix and a matrix. If the eigenvalues of are and , including algebraic multiplicity, what are the eigenvalues of ?

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

56 An invertible matrix has eigenvalues , , and . What are the eigenvalues of ?

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

57 An invertible matrix has characteristic polynomial . What is ?

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

58 Let Using the Cayley-Hamilton theorem, which matrix equals ?

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

59 A square matrix satisfies . Which expression equals ?

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

60 An invertible matrix has characteristic polynomial . Which expression for follows from the Cayley-Hamilton theorem?

Cayley-Hamilton theorem Hard
A.
B.
C.
D.
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Unit 2