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JEE (Main) Mathematics - MATRICES AND DETERMINANTS MCQs

JEE (Main) Mathematics - MATRICES AND DETERMINANTS MCQs (61 to 75)

61. If \( A \) is a square matrix and \( \text{adj}(A) \) is its adjoint, then which of the following is true?

a) \( A \times \text{adj}(A) = I \)

b) \( A \times \text{adj}(A) = A \)

c) \( \text{adj}(A) \times A = A \)

d) \( \text{adj}(A) \times A = I \)

Answer: a) \( A \times \text{adj}(A) = I \)

62. If the determinant of matrix \( A \) is 5, then the determinant of \( 3A \) is:

a) 5

b) 15

c) 125

d) 45

Answer: b) 15

63. The rank of a matrix is defined as:

a) The number of non-zero rows in the echelon form

b) The number of non-zero columns in the echelon form

c) The maximum number of linearly independent rows or columns

d) None of the above

Answer: c) The maximum number of linearly independent rows or columns

64. The matrix \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \) has:

a) Determinant = -2

b) Determinant = 2

c) Determinant = 0

d) Determinant = -4

Answer: a) Determinant = -2

65. For the system of equations \( x + y = 5 \) and \( 2x + 3y = 8 \), the coefficient matrix is:

a) \( \begin{bmatrix} 1 & 2 \\ 1 & 3 \end{bmatrix} \)

b) \( \begin{bmatrix} 1 & 1 \\ 2 & 3 \end{bmatrix} \)

c) \( \begin{bmatrix} 1 & 1 \\ 3 & 2 \end{bmatrix} \)

d) \( \begin{bmatrix} 2 & 3 \\ 1 & 1 \end{bmatrix} \)

Answer: b) \( \begin{bmatrix} 1 & 1 \\ 2 & 3 \end{bmatrix} \)

66. The inverse of matrix \( A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \) is:

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

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

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

d) None of the above

Answer: b) \( \frac{1}{-2} \begin{bmatrix} -4 & 2 \\ 3 & -1 \end{bmatrix} \)

67. The area of a triangle with vertices \( A(1,2) \), \( B(3,4) \), and \( C(5,6) \) can be found using:

a) \( \frac{1}{2} \times \text{det} \begin{bmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{bmatrix} \)

b) \( \text{det} \begin{bmatrix} x_1 & y_1 \\ x_2 & y_2 \\ x_3 & y_3 \end{bmatrix} \)

c) \( \frac{1}{2} \times \text{det} \begin{bmatrix} x_1 & y_1 \\ x_2 & y_2 \end{bmatrix} \)

d) None of the above

Answer: a) \( \frac{1}{2} \times \text{det} \begin{bmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{bmatrix} \)

68. If the determinant of a matrix \( A \) is zero, then:

a) The matrix is singular

b) The matrix is non-singular

c) The matrix is invertible

d) The matrix is orthogonal

Answer: a) The matrix is singular

69. The matrix \( \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \) has a determinant of:

a) 0

b) 1

c) -1

d) 6

Answer: a) 0

70. The matrix \( A = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \) is:

a) A diagonal matrix

b) A symmetric matrix

c) An identity matrix

d) None of the above

Answer: c) An identity matrix

71. The inverse of the matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \) is given by:

a) \( \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \)

b) \( \frac{1}{ad+bc} \begin{bmatrix} -d & b \\ c & -a \end{bmatrix} \)

c) \( \frac{1}{ad-bc} \begin{bmatrix} -d & b \\ c & -a \end{bmatrix} \)

d) \( \frac{1}{ad+bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \)

Answer: a) \( \frac{1}{ad-bc} \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \)

72. The determinant of the matrix \( \begin{bmatrix} 2 & 3 \\ 4 & 5 \end{bmatrix} \) is:

a) -2

b) -1

c) 1

d) 2

Answer: a) -2

73. The system of equations \( x + 2y + 3z = 9 \), \( 2x + 3y + 4z = 10 \), and \( 3x + 4y + 5z = 11 \) is consistent if the determinant of the coefficient matrix is:

a) 1

b) 0

c) -1

d) 2

Answer: b) 0

74. If matrix \( A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), the adjoint of \( A \) is:

a) \( \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \)

b) \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \)

c) \( \begin{bmatrix} -a & b \\ c & -d \end{bmatrix} \)

d) \( \begin{bmatrix} a & -b \\ -c & d \end{bmatrix} \)

Answer: a) \( \begin{bmatrix} d & -b \\ -c & a \end{bmatrix} \)

75. The determinant of a 3x3 matrix \( \begin{bmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{bmatrix} \) is:

a) 6

b) 0

c) 1

d) -6

Answer: b) 0

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JEE (Main) Mathematics - MATRICES AND DETERMINANTS MCQs (61 to 75)

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