Linear Algebra (MTH501)

1. If x + 2 is a factor of the characteristic polynomial of matrix C then an eigenvalue of C is

   1. 2
   2. -2
   3. 1/2
   4. 0

2. Which of the following will be the Linear Combination corresponding to \( \begin{pmatrix} -2 & 3 \\ 5 & 1 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} \)?

   1. \( \begin{pmatrix} -2 \\ 3 \\ \end{pmatrix} x + \begin{pmatrix} 5 \\ 1 \\ \end{pmatrix} y \)
   2. \( \begin{pmatrix} 3 \\ 1 \\ \end{pmatrix} x + \begin{pmatrix} -2 \\ 5 \\ \end{pmatrix} y \)
   3. \( \begin{pmatrix} 5 \\ 1 \\ \end{pmatrix} x + \begin{pmatrix} -2 \\ 3 \\ \end{pmatrix} y \)
   4. \( \begin{pmatrix} -2 \\ 5 \\ \end{pmatrix} x + \begin{pmatrix} 3 \\ 1 \\ \end{pmatrix} y \)

3. A system of linear equations is said to be homogeneous if it can be written in the form ________.

   1. AX=B
   2. AX=0

4. Which of the following will be the Matrix Product corresponding to Linear Combination: \( \begin{pmatrix} -2 \\ 5 \\ \end{pmatrix} x + \begin{pmatrix} 3 \\ 1 \\ \end{pmatrix} y \)?

   1. \( \begin{pmatrix} 1 & -3 \\ -5 & -2 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} \)
   2. \( \begin{pmatrix} -2 & 5 \\ 3 & 1 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} \)
   3. \( \begin{pmatrix} -2 & 3 \\ 5 & 1 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} \)
   4. \( \begin{pmatrix} 3 & -2 \\ 1 & 5 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} \)

5. Why inverse of the matrix A= [1 2] is NOT possible?

   1. Because it is a saquare matrix.
   2. Because it is a zero matrix.
   3. Because it is an identity matrix,
   4. Because it is rectahular matrix.

6. If the determinant of the matrix \( A = \begin{bmatrix} 4 & 3 & 5 \\ 3 & 1 & 1 \\ 5 & 7 & 7 \\ \end{bmatrix} \) is 32 and the matrix B is obtained by multiplying any row of A with an integer value 4, then which of the following is true for the matrix B?

   1. Its determinant is 18.
   2. Its determinant is -32.
   3. Its determinant is 128.
   4. The information is not sufficient to calculate the determinant.

7. An \(n × n\) real matrix is invertible if and only if the span of the rows of A is \( R^n \)

   1. True
   2. False

8. If \( A = \begin{bmatrix} 2 & 3 & 5 \\ 0 & 3 & 6 \\ 0 & 0 & 4 \\ \end{bmatrix} \), then which of the following is the value of det(A)?

   1. 6
   2. 18
   3. 24
   4. 36

9. If the equation: \( \begin{pmatrix} -2 & 3 \\ 5 & 1 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} = \begin{pmatrix} b_1 \\ b_2 \\ \end{pmatrix} \) has the solution for all \(b_1, b_2 \in R \), then \( \begin{pmatrix} -2 \\ 5 \\ \end{pmatrix} \) and \( \begin{pmatrix} 3 \\ 1 \\ \end{pmatrix} \) will span ________.

   1. \( R^2 \) space
   2. \( R^3 \) space
   3. R space
   4. Nothing

10. Gauss-Seidel method is also termed as a method of

    1. Elimination Method
    2. False Position Method
    3. Successive Displacement
    4. Iteration Method

11. If a system of equations is solved using the Gauss-Seidel method, then which of the following is NOT true about the matrix M that is derived from the coefficient matrix?

    1. All of its entries below the diagonal must be zero.
    2. All of its entries above the diagonal must be zero.
    3. Its determinant is non-zero.
    4. It is an invertible matrix.

12. If A be the standard matrix of linear transformation \(T : R^n \rightarrow R^m \), then which of the following is true for the mapping from \(R^n \) onto \(R^m \) ?

    1. The columns of A span \( R^n \).
    2. The columns of A span \( R^m \).
    3. The columns of A are linearly independent.
    4. The columns of A are identical.

13. Two simultaneous linear equations in two variables have no solution if their corresponding lines are ________.

    1. parallel and distinct
    2. intersecting
    3. coincident
    4. perpendicular

14. If T be a transformation, then which of the following is true for its linearity?

    1. \( T(cu^r \, gdv^r) = cT(u^r) gd T(v^r) ; \;\;\;\; \) whre 'c' and 'd' are scalars
    2. \( T(cu^r + dv^r) = cT(u^r) + dT(v^r); \;\;\;\; \) whre 'c' and 'd' are scalars
    3. \( T(cu^r × dv^r) = cT(u^r) × dT(v^r); \;\;\;\; \) whre 'c' and 'd' are scalars
    4. \( T(cu^r + dv^r) = dT(u^r) + cT(v^r); \;\;\;\; \) whre 'c' and 'd' are scalars

15. Let A and B be the square matrices. Then A and B are invertible with \(B = A^{-1} \) and \(A = B^{-1} \) if and only if \(AB = BA \) equals to a (an) ________ matrix.

    1. Singular
    2. Square
    3. Identity
    4. Rectangular

16. How many Pivot partitions the matrix: \( \begin{pmatrix} 2 & 3 & 1 \\ 4 & 6 & 2 \\ \end{pmatrix} \) will have?

    1. 1
    2. 2
    3. 3
    4. 4

17. If \( v_1^r = (2, 1), v_2^r = (3, 4) \) and \( v_3^r = (7, 8) \) then which of the following is true?

    1. \( \{v_1^r, v_2^r, v_3^r\} \) is linearly dependent.
    2. \( \{v_1^r, v_2^r, v_3^r\} \) is linearly independent.
    3. The vector equation has trivial solution.
    4. \( v_1^r = {2 \over 3} v_2^r \)

18. A square matrix A is said to be diagonal if A is similar to a matrix

    1. Column matrix
    2. Zero matrix
    3. Diagonal matrix
    4. None of the given

19. If one of the eigenvalues of \( \begin{bmatrix} A \end{bmatrix} _{n×n} \) is zero, it implies ________

    1. The solution to \( \begin{bmatrix} A \end{bmatrix} \begin{bmatrix} X \end{bmatrix} = \begin{bmatrix} C \end{bmatrix}\) a system of equations is unique
    2. The determinant of \( \begin{bmatrix} A \end{bmatrix} \) is zero
    3. The solution to \( \begin{bmatrix} A \end{bmatrix} \begin{bmatrix} X \end{bmatrix} = \begin{bmatrix} 0 \end{bmatrix}\) system of equations is trivial
    4. The determinant of \( \begin{bmatrix} A \end{bmatrix} \) is nonzero

20. If \(v_1^r , v_2^r \) and \(v_3^r \) are in \(R^m \) then which of the following is equivalent to \(\begin{bmatrix} v_1 & v_2 & v_3 \\ \end{bmatrix} \begin{bmatrix} 2 \\ -7 \\ 5 \\ \end{bmatrix} \)

    1. \(2v_1^r - 7v_2^r + 5v_3^r \)
    2. \(5v_1^r - 7v_2^r + 2v_3^r \)
    3. \(5v_1^r + 2v_2^r - 7v_3^r \)
    4. \(2v_1^r + 5v_2^r - 7v_3^r \)

21. What is the maximum possiblle number of pivots in a 6 × 6 matrix?

    1. 0
    2. 2
    3. 4
    4. 6

22. Which one of the following is a null matrix?

    1. \( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \)
    2. \( \begin{bmatrix} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \end{bmatrix} \)

23. What is Eigen value?

    1. A vector obtained from the coordinates
    2. A matrix determined from the algebraic equations
    3. A scalar associated with a given linear transformation

24. Which of the following is the most appropriate operation(s) for the linear transformation

    1. Scalar multiplication
    2. Vector addition and scalar multiplication
    3. Vector addition
    4. Vector and scalar multiplications

25. A system of linear equations is said to be homogeneous if the constant terms are all

    1. One
    2. Zero
    3. Both (a) and (b)
    4. None of the above

26. In A is a square matrix, then the minor of entry ith row and jth column is to be the determinant of the sub matrix that remains when the ith row and jth column of A are

    1. added
    2. deleted
    3. multiplied
    4. divided

27. The Elementary Row operations: \(R'_2 \rightarrow R_2 + 4R_1 \) and \(R'_3 \rightarrow R_3 - 6R_1 \) are performed on to get \( \begin{pmatrix} 1 & 2 & -5 \\ -4 & 1 & -6 \\ 6 & 3 & -4 \end{pmatrix} \sim \) ________?

    1. \( \begin{pmatrix} 1 & 2 & -5 \\ 0 & -9 & -26 \\ 0 & -9 & 26 \end{pmatrix} \)
    2. \( \begin{pmatrix} 1 & 2 & -5 \\ 0 & 9 & 26 \\ 0 & -9 & -26 \end{pmatrix} \)
    3. \( \begin{pmatrix} 1 & 2 & -5 \\ 0 & 9 & -26 \\ 0 & -9 & 26 \end{pmatrix} \)
    4. \( \begin{pmatrix} 1 & 2 & -5 \\ 0 & -9 & 26 \\ 0 & 9 & -26 \end{pmatrix} \)

28. If \( A = \begin{bmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix} \), then which of the following is true for A?

    1. det(A) = 1
    2. det(A) = - 1
    3. det(A) = 0
    4. det(A) = ±1

29. If u + v = u + w, then:

    1. v + w
    2. v ≠ w
    3. v = w
    4. None of the given

30. A 3 × 3 identity matrix have three and ________ eigen values.

    1. same
    2. distinct

31. If \( Ax^r = b^r \) and factorization of A is LU, then which of the following pair of equations can be used to solve \( LUx^r = b^r \) for value of '\( x^r \)'?

    1. \( Ux^r = y^r \) and \( Ly^r = b^r \)
    2. \( Lx^r = y^r \) and \( Uy^r = b^r \)
    3. \( Ub^r = y^r \) and \( Ly^r = x^r \)
    4. \( Lb^r = y^r \) and \( Uy^r = x^r \)

32. Let A be the matrix of order 2x3 and B be the matrix of order 3x5, then which of the following is the order of the matrix AB?

    1. 2x3
    2. 3x5
    3. 3x3
    4. 2x5

33. Let V be a five-dimensional vector space, and let S be a subset of V which spans V. Then S

    1. Must be linearly dependent
    2. Must be a basis for V
    3. Must have infinitely many elements
    4. Must have at most five elements

34. A sufficient condition for the jacobi's method to converge for the linear system Ax=b

    1. A is diagonally dominant
    2. A-I is diagonally dominant
    3. A is non-singular
    4. None of the given

35. If \(v_1 = (2, 2, 2), v_2 = (0, 0, 3) \) and \(v_3 = (0, 1, 1) \) span \( R^3 \), then which of the following is true for any arbitrary \(b^r = (b_1, b_2, b_3) \in R^3 \) ?

    1. \( (0, 1, 1) = k_1 (b_1, b_2, b_3) + k_2 (2, 2, 2) + k_3 (0, 0, 3) \)
    2. \( (b_1, b_2, b_3) = k_1 (2, 2, 2) + k_2 (0, 0, 3) + k_3 (0, 1, 1) \)
    3. \( (0, 0, 3) = k_1 (2, 2, 2) + k_2 (b_1, b_2, b_3) + k_3 (0, 1, 1) \)
    4. \( (0, 1, 1) = k_1 (2, 2, 2) + k_2 (0, 0, 3) + k_3 (b_1, b_2, b_3) \)

36. Which of the following is true about the existence of free variables (parameter) in a system of linear equations?

    1. They guarantee the Consistency.
    2. They guarantee the Inconsistency.
    3. They do not guarantee the Consistency.
    4. None of the given.

37. If \( X =\begin{bmatrix} M \\ N \\ \end{bmatrix} \) and \( Y= \begin{bmatrix} Q & P \\ \end{bmatrix} \) (Whare \(\mathbf{M, N, Q}\) and \(\mathbf{P}\) are saqure sub-matrices of same size), then Which of the following is possible?

    1. The product \(\mathbf{XY}\) and \(\mathbf{YX}\) both are not defined
    2. The product \(\mathbf{XY}\) and \(\mathbf{YX}\) both are defined
    3. The product \(\mathbf{XY}\) is defied but \(\mathbf{YX}\) is not defined
    4. None of the given

38. Let A be n × n matrix, then A is invertible if and only if

    1. det A is not zero
    2. det A is zero

39. If \( A = \begin{bmatrix} 2 & 1 \\ 4 & 3 \\ \end{bmatrix} \) and \( B = \begin{bmatrix} 1+1 & 2-1 \\ 2+2 & 4-1 \\ \end{bmatrix} \), then which of the following is true for A and B?

    1. A and B are equal matrices.
    2. A is the transpose of B.
    3. B is the transpose of A.
    4. B is the multiplicative inverse of A.

40. If λ + 2 is a factor of the characteristic polynomial of matrix C, then which of the following is the Eigenvalue of C?

    1. 2
    2. -2
    3. \(1 \over 2\)
    4. 0

41. If AB = I = BA for matrices A, B and I, where I is an identity matrix, then

    1. B is inverse of A
    2. A is inverse of B
    3. A^(-1) = B, B^(-1) = A,
    4. All of the given

42. If λ is an eigenvector of A, then every nonzero vector x such that Ax = λx is called an ________ of A corresponding to ________.

    1. Eigenvalue, λ
    2. Eigenvector, λ
    3. Eigenvalue, A
    4. Eigenvector, A

43. If \(M = \begin{bmatrix} 3 \\ \end{bmatrix} \) then Which of the following is the determinant of the matrix M?

    1. 1
    2. [1]
    3. 3
    4. [3]

44. 7x is an algebraic term in which 7 is a ________ and x is a ________.

    1. term, expression
    2. coefficient, variable
    3. variable, coefficient
    4. numerical, alphabet

45. \(9x^2 + 3x + 4 \) is ________.

    1. an equation
    2. a term
    3. an algebraic expression
    4. quadratic equation

46. Let a matrix A has both negative and positive eigen values, so in this case origin behaves as a ________ point.

    1. Saddle
    2. Critical

47. The Invertible Matrix Theorem applies only to ________ matrices.

    1. Rectangular
    2. Square
    3. Identity
    4. Scalar

48. If the equation: \( \begin{pmatrix} -2 & 3 \\ 5 & 1 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} = \begin{pmatrix} b_1 \\ b_2 \\ \end{pmatrix} \) has the solution for all \(b_1, b_2 \in R \), then \( \begin{pmatrix} b_1 \\ b_2 \\ \end{pmatrix} \in \) ________.

    1. Span\( \begin{Bmatrix} \begin{pmatrix} -2 \\ 3 \\ \end{pmatrix} , \begin{pmatrix} 5 \\ 1 \\ \end{pmatrix} \end{Bmatrix} \)
    2. Span\( \begin{Bmatrix} \begin{pmatrix} -2 \\ 1 \\ \end{pmatrix} , \begin{pmatrix} 5 \\ 3 \\ \end{pmatrix} \end{Bmatrix} \)
    3. Span\( \begin{Bmatrix} \begin{pmatrix} 3 \\ -2 \\ \end{pmatrix} , \begin{pmatrix} 1 \\ 5 \\ \end{pmatrix} \end{Bmatrix} \)
    4. Span\( \begin{Bmatrix} \begin{pmatrix} -2 \\ 5 \\ \end{pmatrix} , \begin{pmatrix} 3 \\ 1 \\ \end{pmatrix} \end{Bmatrix} \)

49. Which of the following Elementary Row operations would perform in order to get \( \begin{pmatrix} 1 & 2 & -5 \\ -4 & 1 & -6 \\ 6 & 3 & -4 \\ \end{pmatrix} \sim \begin{pmatrix} 1 & 2 & -5 \\ 0 & 9 & -26 \\ 0 & -9 & 26 \\ \end{pmatrix} \)?

    1. \(R'_2 \rightarrow R_2 - 4R_1, R'_3 \rightarrow R_3 + 6R_1 \)
    2. \(R'_2 \rightarrow R_2 + 4R_1, R'_3 \rightarrow R_3 - 6R_1 \)
    3. \(R'_2 \rightarrow R_1 + 4R_2, R'_3 \rightarrow R_1 - 6R_3 \)
    4. \(R'_2 \rightarrow R_1 - 4R_2, R'_3 \rightarrow R_1 + 6R_3 \)

50. If the determinant of the matrix \( A = \begin{bmatrix} 2 & 3 & 4 \\ 1 & 2 & 2 \\ 3 & 4 & 5 \\ \end{bmatrix} \) is -1 and the matrix B is obtained by adding 2 times of the second row in the first row of the matrix A, then which of the following is true for the matrix B?

    1. Its determinant is -1.
    2. Its determinant is 1.
    3. Its determinant can not be evaluated.
    4. The information is not sufficient to calculate the determinant.

51. Which of the following is true for the matrix \( \begin{bmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3\\ \end{bmatrix} \)?

    1. It is a null matrix.
    2. It is a scalar matrix.
    3. It is a diagonal matrix.
    4. It is an identity matrix.

52. Which of the following is the simplified form of \(-1 \begin{bmatrix} -1 & 2 \\ \end{bmatrix} + \begin{bmatrix} 2 & 3 \\ \end{bmatrix} \)

    1. \( \begin{bmatrix} 3 & -1 \\ \end{bmatrix} \)
    2. \( \begin{bmatrix} -3 & 1 \\ \end{bmatrix} \)
    3. \( \begin{bmatrix} 3 & 1 \\ \end{bmatrix} \)
    4. \( \begin{bmatrix} -3 & -1 \\ \end{bmatrix} \)

53. Which of the following is the coefficient matrix for the system \( \begin{matrix} x_1 - 2x_2 + x_3 = 0 \\ 2x_2 - 7x_3 = 8 \\ -4x_1 + 3x_2 + 9x_3 = -6 \end{matrix} \)

    1. \( \begin{bmatrix} 1 & -2 & 1 \\ 0 & 2 & -7 \\ -4 & 3 & 9 \\ \end{bmatrix} \)
    2. \( \begin{bmatrix} 1 & -2 & 0 \\ 0 & 2 & 8 \\ -4 & 3 & -6 \\ \end{bmatrix} \)
    3. \( \begin{bmatrix} 1 & 1 & 0 \\ 0 & -7 & 8 \\ -4 & 9 & -6 \\ \end{bmatrix} \)
    4. \( \begin{bmatrix} 1 & 0 & -4 \\ -2 & 2 & 3 \\ 1 & -7 & 9 \\ \end{bmatrix} \)

54. A homogeneous linear system always has the trivial solution: there are only two possibilities for its solutions:

    1. The system has only the trivial solution
    2. The system has infinitely many solutionsnin addition to trivial solution
    3. Both (a) and (b)
    4. None of the above

55. If \( A = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & k & 1 \\ \end{bmatrix} \), then which of the following is true for the matrix?

    1. det(A) = 1
    2. det(A) = k - 1
    3. det(A) = k
    4. det(A) = k + 1

56. Since every linear transformation \(T : R^n \rightarrow R^m \) is actually matrix transformation, then which of the following is the alternate notation for the transformation?

    1. \(Ax^r \; a \;\; x^r \)
    2. \(Ax^r \; a \;\; T(x^r) \)
    3. \(x^r \; a \;\; Ax^r \)
    4. \(T(x^r) \; a \;\; Ax^r \)

57. Let 'Ax = 0' be a homogeneous linear system of 'n' equations and 'n' unknowns. Then, the coefficient matrix 'A' is invertible if and only if this system has ________ solution.

    1. No
    2. trivial
    3. non-trivial
    4. infinite many

58. The equation: \( \begin{pmatrix} -2 & 3 \\ 5 & 1 \\ \end{pmatrix} \begin{pmatrix} x \\ y \\ \end{pmatrix} = \begin{pmatrix} -7 \\ 4 \\ \end{pmatrix} \) will have the solution only if ________.

    1. \( \begin{pmatrix} -7 \\ 4 \\ \end{pmatrix} = x \begin{pmatrix} 3 \\ 1 \\ \end{pmatrix} + y \begin{pmatrix} -2 \\ 5 \\ \end{pmatrix} \)
    2. \( \begin{pmatrix} -7 \\ 4 \\ \end{pmatrix} = x \begin{pmatrix} -2 \\ 5 \\ \end{pmatrix} + y \begin{pmatrix} 3 \\ 1 \\ \end{pmatrix} \)
    3. \( \begin{pmatrix} -7 \\ 4 \\ \end{pmatrix} = x \begin{pmatrix} -2 \\ 3 \\ \end{pmatrix} + y \begin{pmatrix} 5 \\ 1 \\ \end{pmatrix} \)
    4. \( \begin{pmatrix} -7 \\ 4 \\ \end{pmatrix} = x \begin{pmatrix} 5 \\ 1 \\ \end{pmatrix} + y \begin{pmatrix} -2 \\ 3 \\ \end{pmatrix} \)

59. An n × n matrix A is said to be diagonalizable if and only if A has n ________ eigenvectors.

    1. Linearly dependent
    2. Linearly Independent

60. If a homogeneous system \(Ax = 0 \) has a trivial solution, then which of the following is(are) the value(s) of the vector x?

    1. -1
    2. 0
    3. 1
    4. 2

61. The solution of Ax = b exists if and only if b can be written as a linear combination of ________ of A.

    1. columns
    2. rows
    3. both columns and rows
    4. elements

62. If x - 2 is a factor of the characteristic polynomial of matrix C then an eigenvalue of C is

    1. 2
    2. -2
    3. 1/2
    4. 0

Select a course code for Objective Questions:

ACC311 ACC501 BIF101 BIF401 BIF501 BIO101 BIO201 BIO202 BIO203 BIO204 BIO301 BIO302 BIO303 BIO401 BIO503 BNK603 BT101 BT201 BT301 BT302 BT401 BT402 BT403 BT404 BT405 BT406 BT501 BT504 BT505 BT604 CHE201 CHE301 CS001 CS101 CS201 CS202 CS204 CS301 CS302 CS304 CS311 CS401 CS402 CS403 CS408 CS501 CS502 CS504 CS506 CS508 CS601 CS602 CS604 CS605 CS607 CS609 CS610 CS614 CS707 CS710 ECO401 ECO402 ECO403 ECO404 EDU302 EDU401 EDU402 EDU501 EDU601 ENG101 ENG201 ENG301 ENG501 FIN611 FIN624 FIN625 ISL201 IT430 MCM101 MCM301 MGMT623 MGT101 MGT211 MGT301 MGT401 MGT402 MGT501 MGT502 MGT503 MGT510 MGT601 MGT603 MKT501 MKT610 MKT630 MTH101 MTH202 MTH302 MTH501 MTH601 MTH634 PAK301 PHY101 PHY301 PSY101 PSY403 PSY502 PSY512 SOC101 SOC301 SOC401 STA301 STA630 STA641 ZOO102 ZOO401 ZOO501 ZOO502 ZOO503 ZOO504 ZOO505 ZOO506 ZOO510 ZOO630

Select a course code for Subjective Questions:

ACC311 CS001 CS101 CS201 CS301 CS607 CS701 CS702 CS703 CS704 CS707 CS708 CS710 CS711 CS718 CS721 CS724 ECO401 ECO403 ENG101 ENG301 MGT703 MTH501 STA301