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Unit 1 - Practice Quiz

MTH165

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

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

A. Adding a multiple of one row to another ()
B. Interchanging two rows ()
C. Multiplying a row by 1
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. Zero
B. Not zero
C. Positive
D. Negative

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

A. $0$
B. $1$
C.
D.

8 Find the rank of the matrix

A.
B. 1
C. 2
D. 4

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

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

10 The rank of a matrix is equal to:

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

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

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

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

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. Infinitely many solutions
D. Only the trivial solution

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

A. No solution
B. A unique solution
C. Infinitely many solutions
D. Exactly two 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. At least one solution (Trivial solution)
C. Only non-trivial solutions
D. Infinite solutions

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

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

21 Consider the system: and . This system is:

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

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

A.
B.
C.
D. Any real number

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. Eigenvalues
C. Latent vectors
D. Rank

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

A.
B.
C.
D.

30 Find the eigenvalues of the matrix

A. 1, 2
B. 1, 3
C. 0, 3
D. 2, 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. Trace of the matrix
B. Determinant of the matrix
C. Rank of the matrix
D. 1

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

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

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

36 The eigenvalues of a skew-symmetric matrix are:

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

37 At least one eigenvalue of a singular matrix is:

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

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

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

39 What are the eigenvalues of the matrix ?

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

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

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

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

43 Given , the inverse is given by:

A.
B.
C.
D.

44 An eigenvector corresponding to an eigenvalue must be:

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

45 Algebraic Multiplicity of an eigenvalue refers to:

A. The number of linearly independent eigenvectors associated with it
B. The number of times the eigenvalue appears as a root of the characteristic equation
C. The value of the eigenvalue itself
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. 1, 0
C. 0, 0
D. -1, -1

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. -2
C.
D. 3

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

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