Back to Subject

Unit 1: Matrix Algebra - Practice Quiz

MTH174 — Engineering Mathematics 60 Questions
0 Correct 0 Wrong 60 Left
0/60

1 Which operation is an elementary row operation?

Elementary operations and their use in getting the rank of a matrix Easy
A. Adding two columns only
B. Interchanging two rows
C. Deleting a row
D. Squaring a row

2 The rank of a matrix is the maximum number of linearly independent:

Elementary operations and their use in getting the rank of a matrix Easy
A. Diagonal entries
B. Rows or columns
C. Zero entries
D. Eigenvalues only

3 What is the rank of the zero matrix of order ?

Elementary operations and their use in getting the rank of a matrix Easy
A.
B.
C.
D.

4 Which elementary operation does not change the rank of a matrix?

Elementary operations and their use in getting the rank of a matrix Easy
A. Multiplying a row by a nonzero number
B. Replacing a row by zero
C. Deleting an independent row
D. Changing every entry to its square

5 A square matrix has an inverse if and only if:

Inverse of a matrix Easy
A.
B. has a zero row
C.
D. is rectangular

6 If is the inverse of , then equals:

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

7 What is the inverse of the identity matrix ?

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

8 For , the determinant of is:

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

9 If is a square matrix and exists, then is called:

Inverse of a matrix Easy
A. Nonsingular
B. Rectangular
C. Zero
D. Singular

10 The equations and have the solution:

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

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

Solution of linear simultaneous equations Easy
A. More variables than equations
B. Only zero coefficients
C. At least one solution
D. No solution

12 The matrix form of a system of linear equations is commonly written as:

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

13 If the coefficient matrix is invertible in , the solution is:

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

14 The equations and represent:

Solution of linear simultaneous equations Easy
A. A single point
B. Two parallel lines
C. The same line
D. Two perpendicular lines

15 A scalar is an eigenvalue of if there is a nonzero vector such that:

Eigen-values and eigenvectors of a matrix Easy
A.
B. for every
C.
D.

16 An eigenvector of a matrix must be:

Eigen-values and eigenvectors of a matrix Easy
A. A determinant
B. A nonzero vector
C. A zero vector
D. A scalar only

17 The eigenvalues of a matrix are found from the equation:

Eigen-values and eigenvectors of a matrix Easy
A.
B.
C.
D.

18 The eigenvalues of a diagonal matrix are its:

Eigen-values and eigenvectors of a matrix Easy
A. Column products
B. Row sums
C. Off-diagonal entries
D. Diagonal entries

19 If and is an eigenvector, then cannot be:

Eigen-values and eigenvectors of a matrix Easy
A. An eigenvector
B. A nonzero vector
C. The zero vector
D. A column vector

20 The Cayley-Hamilton theorem states that every square matrix satisfies its own:

Cayley-Hamilton theorem Easy
A. Characteristic equation
B. Linear equation
C. Row equation
D. Inverse equation

21 Find the rank of the matrix .

Elementary operations and their use in getting the rank of a matrix Medium
A.
B.
C.
D.

22 For , what is the rank when ?

Elementary operations and their use in getting the rank of a matrix Medium
A.
B.
C.
D.

23 Which elementary row operation leaves the rank of a matrix unchanged?

Elementary operations and their use in getting the rank of a matrix Medium
A. Replacing a row by its square
B. Deleting a nonzero row
C. Multiplying a row by zero
D. Adding a multiple of one row to another

24 The echelon form of a matrix is . What is its rank?

Elementary operations and their use in getting the rank of a matrix Medium
A.
B.
C.
D.

25 Find the inverse of .

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

26 For which value of does the matrix fail to have an inverse?

Inverse of a matrix Medium
A.
B.
C.
D. only

27 If is a nonsingular matrix and , then which statement is necessarily true?

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

28 Find the inverse of .

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

29 Solve the system and .

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

30 Determine the nature of the solutions of , , and .

Solution of linear simultaneous equations Medium
A. A unique solution
B. Exactly two solutions
C. No solution
D. Infinitely many solutions

31 For the system and , when does the system have infinitely many solutions?

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

32 Using Cramer's rule, find in the system and .

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

33 Find the eigenvalues of .

Eigen-values and eigenvectors of a matrix Medium
A. and
B. and
C. and
D. and

34 An eigenvector corresponding to the eigenvalue of is:

Eigen-values and eigenvectors of a matrix Medium
A.
B.
C.
D.

35 If the eigenvalues of a matrix are , , and , what is its trace?

Eigen-values and eigenvectors of a matrix Medium
A.
B.
C.
D.

36 If is an eigenvalue of an invertible matrix , which is an eigenvalue of ?

Eigen-values and eigenvectors of a matrix Medium
A.
B.
C.
D.

37 For , what is the algebraic multiplicity of its eigenvalue?

Eigen-values and eigenvectors of a matrix Medium
A. The eigenvalue has multiplicity
B. The eigenvalue has multiplicity
C. The eigenvalue has multiplicity
D. The eigenvalue has multiplicity

38 The characteristic polynomial of is . By the Cayley-Hamilton theorem, which equation is satisfied by ?

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

39 If a matrix satisfies , express in terms of and .

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

40 If , find in terms of and .

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

41 For the matrix , determine its rank according to the value of .

Elementary operations and their use in getting the rank of a matrix Hard
A. for all
B. for all
C. if , and otherwise
D. if , and otherwise

42 Let be a matrix whose reduced row-echelon form is . Which statement is necessarily true?

Elementary operations and their use in getting the rank of a matrix Hard
A. The fourth column is a pivot column of the original matrix
B. The nullity of is
C. Every solution of has its third component equal to zero
D. The column space of has dimension

43 For , find the values of for which .

Elementary operations and their use in getting the rank of a matrix Hard
A.
B.
C.
D.

44 Which operation can change the numerical values of minors but cannot change the rank of a matrix?

Elementary operations and their use in getting the rank of a matrix Hard
A. Deleting a nonzero row
B. Multiplying a row by a nonzero scalar
C. Multiplying a row by the zero scalar
D. Replacing a row by a non-equivalent linear combination

45 The inverse of is

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

46 For , which matrix satisfies ?

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

47 Suppose a square matrix satisfies . Which expression gives ?

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

48 Let be an matrix. If , which conclusion is always valid?

Inverse of a matrix Hard
A.
B. is necessarily singular
C. has no eigenvalue equal to
D.

49 Consider the system , , and . Which condition gives infinitely many solutions?

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

50 For the system , , and , the solution set is

Solution of linear simultaneous equations Hard
A. A two-parameter family of solutions
B. No solution
C. A one-parameter family of solutions
D. A unique solution

51 For the system , , and , determine the value of for consistency.

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

52 If the coefficient matrix of a system has rank and the augmented matrix has rank , then the system has

Solution of linear simultaneous equations Hard
A. Exactly two solutions
B. Infinitely many solutions
C. No solution
D. A unique solution

53 For , which pair correctly gives an eigenvalue and a corresponding eigenvector?

Eigen-values and eigenvectors of a matrix Hard
A. ,
B. ,
C. ,
D. ,

54 For , the eigenspace corresponding to is

Eigen-values and eigenvectors of a matrix Hard
A.
B.
C.
D.

55 The matrix is not diagonalizable because

Eigen-values and eigenvectors of a matrix Hard
A. It has no real eigenvalues
B. Its characteristic polynomial has no repeated root
C. Its eigenspace has dimension less than the algebraic multiplicity
D. Its determinant is equal to zero

56 If a matrix has characteristic polynomial and the eigenspace for has dimension , which statement is correct?

Eigen-values and eigenvectors of a matrix Hard
A. The matrix is diagonalizable
B. The eigenvalue has geometric multiplicity zero
C. The matrix is not diagonalizable
D. The matrix has three linearly independent eigenvectors

57 If is an eigenvector of an invertible matrix with eigenvalue , which is necessarily true?

Eigen-values and eigenvectors of a matrix Hard
A.
B.
C.
D.

58 A matrix has characteristic polynomial . Using the Cayley-Hamilton theorem, express in terms of , , and .

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

59 If satisfies , which formula gives ?

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

60 Let be a matrix with characteristic polynomial . Which identity follows directly from Cayley-Hamilton?

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