AP POLYCET MATHEMATICS Multiple Choice Questions (MCQs)

AP POLYCET Mathematics - Matrices: Multiple Choice Questions

61. What is the order of the matrix \( A = \begin{pmatrix} 3 & 5 & 7 \\ 2 & 4 & 6 \end{pmatrix} \)?
    a) 2 x 3
    b) 3 x 2
    c) 3 x 3
    d) 2 x 2
    Answer: a) 2 x 3
62. If \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \) and \( B = \begin{pmatrix} 5 & 6 \\ 7 & 8 \end{pmatrix} \), what is \( A + B \)?
    a) \( \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix} \)
    b) \( \begin{pmatrix} 6 & 8 \\ 6 & 8 \end{pmatrix} \)
    c) \( \begin{pmatrix} 5 & 7 \\ 10 & 12 \end{pmatrix} \)
    d) \( \begin{pmatrix} 1 & 2 \\ 7 & 8 \end{pmatrix} \)
    Answer: a) \( \begin{pmatrix} 6 & 8 \\ 10 & 12 \end{pmatrix} \)
63. Which of the following matrices is the identity matrix for 2 x 2 matrices?
    a) \( \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \)
    b) \( \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \)
    c) \( \begin{pmatrix} 1 & 1 \\ 0 & 0 \end{pmatrix} \)
    d) \( \begin{pmatrix} 2 & 0 \\ 0 & 2 \end{pmatrix} \)
    Answer: a) \( \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \)
64. What is the determinant of the matrix \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \)?
    a) -2
    b) 2
    c) 4
    d) -4
    Answer: a) -2
65. If \( A \) is a 3 x 3 matrix and \( B \) is a 3 x 3 matrix, which of the following is true about the multiplication \( AB \)?
    a) The product is a 3 x 3 matrix.
    b) The product is a 2 x 2 matrix.
    c) The product is not possible.
    d) The product is a 3 x 2 matrix.
    Answer: a) The product is a 3 x 3 matrix.
66. Which of the following is the inverse of the matrix \( A = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix} \)?
    a) \( \begin{pmatrix} 4 & -3 \\ -1 & 2 \end{pmatrix} \)
    b) \( \begin{pmatrix} -4 & 3 \\ 1 & -2 \end{pmatrix} \)
    c) \( \begin{pmatrix} -2 & 3 \\ 1 & 4 \end{pmatrix} \)
    d) \( \begin{pmatrix} 4 & 3 \\ 1 & -2 \end{pmatrix} \)
    Answer: a) \( \begin{pmatrix} 4 & -3 \\ -1 & 2 \end{pmatrix} \)
67. What is the trace of the matrix \( A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \)?
    a) 15
    b) 12
    c) 9
    d) 5
    Answer: a) 15
68. If the matrix \( A \) is symmetric, what is true about its elements?
    a) \( a_{ij} = a_{ji} \)
    b) \( a_{ij} = -a_{ji} \)
    c) \( a_{ij} = 0 \)
    d) \( a_{ij} \neq a_{ji} \)
    Answer: a) \( a_{ij} = a_{ji} \)
69. Which of the following matrices is a diagonal matrix?
    a) \( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix} \)
    b) \( \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 7 & 8 & 9 \end{pmatrix} \)
    c) \( \begin{pmatrix} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix} \)
    d) \( \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \)
    Answer: a) \( \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix} \)
70. What is the result of multiplying any matrix by the identity matrix?
    a) The matrix itself
    b) A zero matrix
    c) A matrix of ones
    d) A scalar matrix
    Answer: a) The matrix itself
71. Which of the following is not a property of matrix multiplication?
    a) Matrix multiplication is associative.
    b) Matrix multiplication is commutative.
    c) Matrix multiplication is distributive.
    d) Matrix multiplication has an identity element.
    Answer: b) Matrix multiplication is commutative.
72. What is the rank of the matrix \( A = \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \)?
    a) 2
    b) 1
    c) 0
    d) 3
    Answer: b) 1
73. What is the inverse of a matrix \( A \) if \( A \) is a 2 x 2 matrix and its determinant is 0?
    a) The inverse exists.
    b) The inverse does not exist.
    c) The inverse is a zero matrix.
    d) The inverse is the identity matrix.
    Answer: b) The inverse does not exist.
74. Which of the following is a property of the determinant of a matrix?
    a) The determinant of a matrix is always positive.
    b) The determinant of a matrix is always zero.
    c) The determinant is multiplicative.
    d) The determinant is non-zero for all matrices.
    Answer: c) The determinant is multiplicative.
75. Which of the following represents the matrix transpose of \( A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix} \)?
    a) \( \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix} \)
    b) \( \begin{pmatrix} 2 & 4 \\ 1 & 3 \end{pmatrix} \)
    c) \( \begin{pmatrix} 1 & 2 \\ 4 & 3 \end{pmatrix} \)
    d) \( \begin{pmatrix} 3 & 1 \\ 4 & 2 \end{pmatrix} \)
    Answer: a) \( \begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix} \)

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