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Unit 1: Linear Algebra - Practice Quiz

MTH165 — Mathematics For Engineers 50 Questions
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1 If matrix is of order and matrix is of order , then the order of matrix is:

A.
B.
C.
D.

2 Which of the following is true for the transpose of the product of two matrices ?

A.
B.
C.
D.

3 A square matrix is said to be symmetric if:

A.
B.
C.
D.

4 The trace of a matrix is defined as:

A. The product of the diagonal elements
B. The sum of the principal diagonal elements
C. The determinant of the matrix
D. The sum of all elements

5 Which elementary row operation changes the value of the determinant of a matrix?

A. Multiplying a row by 1
B. Adding a multiple of one row to another ()
C. Interchanging two rows ()
D. None of the above

6 The rank of a matrix is defined as the order of the largest square sub-matrix whose determinant is:

A. Not zero
B. Negative
C. Zero
D. Positive

7 What is the rank of the identity matrix of order ?

A.
B.
C.
D.

8 Find the rank of the matrix

A. 4
B. 1
C. 2
D. 0

9 If a matrix of order is singular (), then its rank is:

A. Exactly 0
B. Greater than 3
C. Less than 3
D. Exactly 3

10 The rank of a matrix is equal to:

A. The number of columns
B. The number of non-zero rows in its Echelon form
C. The sum of diagonal elements
D. The number of rows

11 A set of vectors is linearly dependent if there exist scalars (not all zero) such that:

A.
B.
C.
D.

12 Two vectors in , and , are:

A. Orthogonal
B. Linearly Dependent
C. Linearly Independent
D. Orthonormal

13 If the determinant of a matrix formed by vectors as columns is non-zero, then the vectors are:

A. Parallel
B. Linearly Dependent
C. Linearly Independent
D. Zero vectors

14 What is the maximum number of linearly independent vectors in ?

A.
B.
C.
D.

15 A system of linear equations is consistent if and only if:

A.
B.
C.
D.

16 For a system of linear equations in variables , if , then the system has:

A. No solution
B. A unique solution
C. Only the trivial solution
D. Infinitely many solutions

17 For a system of linear equations in variables , if , then the system has:

A. A unique solution
B. No solution
C. Exactly two solutions
D. Infinitely many solutions

18 The system of equations has no solution if:

A.
B.
C.
D. B is a zero vector

19 A homogeneous system of equations always has:

A. No solution
B. Infinite solutions
C. Only non-trivial solutions
D. At least one solution (Trivial solution)

20 A homogeneous system in unknowns has a non-trivial solution if:

A. A is an identity matrix
B.
C.
D.

21 Consider the system: and . This system is:

A. Homogeneous
B. Consistent with unique solution
C. Consistent with infinite solutions
D. Inconsistent

22 For what value of does the system , , have only the trivial solution?

A.
B. Any real number
C.
D.

23 A square matrix is invertible (non-singular) if and only if:

A.
B.
C.
D.

24 The inverse of a matrix is given by:

A.
B.
C.
D.

25 If and are invertible matrices, then is equal to:

A.
B.
C.
D.

26 The inverse of an orthogonal matrix is:

A.
B.
C.
D.

27 If , then is:

A.
B.
C.
D.

28 The roots of the characteristic equation are called:

A. Eigenvectors
B. Latent vectors
C. Rank
D. Eigenvalues

29 If is an eigenvalue of and is the corresponding eigenvector, then:

A.
B.
C.
D.

30 Find the eigenvalues of the matrix

A. 0, 3
B. 1, 2
C. 2, 3
D. 1, 3

31 The sum of the eigenvalues of a matrix is equal to:

A. Determinant of the matrix
B. Trace of the matrix
C. Highest eigenvalue
D. Zero

32 The product of the eigenvalues of a matrix is equal to:

A. 1
B. Rank of the matrix
C. Trace of the matrix
D. Determinant of the matrix

33 If the eigenvalues of a matrix are 2, 3, and 4, then the eigenvalues of are:

A. 4, 9, 16
B. 1/2, 1/3, 1/4
C. 2, 3, 4
D.

34 If is an eigenvalue of a non-singular matrix , then the eigenvalue of is:

A.
B.
C.
D.

35 The eigenvalues of a real symmetric matrix are always:

A. 0
B. Purely imaginary
C. Complex with non-zero imaginary part
D. Real

36 The eigenvalues of a skew-symmetric matrix are:

A. Always positive
B. Always real
C. Always 1
D. Either 0 or purely imaginary

37 At least one eigenvalue of a singular matrix is:

A. -1
B. 0
C. Infinite
D. 1

38 If has eigenvalues 2 and 5, what are the eigenvalues of ?

A. 2, 5
B. 6, 15
C. -1, 2
D. 5, 8

39 What are the eigenvalues of the matrix ?

A. 1, -1
B. 1, 1
C. 0, 0
D. -1, -1

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

A. Transpose
B. Diagonal elements
C. Characteristic equation
D. Inverse

41 If the characteristic equation of a matrix is , then according to Cayley-Hamilton theorem:

A.
B.
C.
D.

42 The Cayley-Hamilton theorem can be used to find:

A. Trace only
B. Inverse of a matrix
C. Rank of a matrix
D. Determinant only

43 Given , the inverse is given by:

A.
B.
C.
D.

44 An eigenvector corresponding to an eigenvalue must be:

A. A unit vector
B. A column of the identity matrix
C. A non-zero vector
D. A zero vector

45 Algebraic Multiplicity of an eigenvalue refers to:

A. The value of the eigenvalue itself
B. The number of times the eigenvalue appears as a root of the characteristic equation
C. The number of linearly independent eigenvectors associated with it
D. The rank of the matrix

46 A square matrix of order is diagonalizable if and only if:

A. It has distinct eigenvalues
B. It has linearly independent eigenvectors
C. Its determinant is non-zero
D. It is symmetric

47 If , its eigenvalues are:

A. 1, 1
B. 0, 0
C. -1, -1
D. 1, 0

48 Which of the following matrices is in Row Echelon Form?

A.
B.
C.
D.

49 If is a matrix with eigenvalues 1, -1, and 2, the determinant of is:

A. -2
B. 0
C. 3
D. 2

50 If is a matrix with eigenvalues 1, -1, and 2, the Trace of is:

A. 3
B. 0
C. -2
D. 2
Next MCQ
Unit 2