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Algebra (multiple choice) questions on matrices

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    I have tried to understand the first few but I can't get the rest, and if you don't understand the questions because i typed them wrong then please tell me!
    Here they are:

    Question 1
    Which of the following is a linear equation/are linear equations in x1, x2 and x3?
    A. 2x1 ? ?3x2 + 0, 032x3 = 5
    B. 7x1x2 ? 4x2 + x3 = 0
    1. Only A 2. Only B 3. Both A and B 4. Neither A nor B.
    I chose 1.

    Question 2
    Which of the following is a solution/are solutions of the system
    x + 2x ? z = 8
    3x ? y + 2z = ?5
    4x + 2y + 3z = ?1
    A. (1, 2, ?3)
    B. (1, 2, 3)
    1. Only A 2. Only B 3. Both A and B 4. Neither A nor B.
    I chose 4

    Question 3
    Suppose
    A =
    1 2 0 0
    0 0 1 0
    0 0 0 1
    Which of the following statements is/are true?
    A. A is in row–echelon form.
    B. A is in reduced row–echelon form.
    1. Only A 2. Only B 3. Both A and B 4. Neither A nor B.
    I chose 2 here.

    Question 4
    Consider the system
    3x + ky = 3
    2x + 2y = 5.
    The system will have a unique solution when k is
    A. not equal to 3.
    B. any real number.
    1. A only 2. B only 3. Both A and B 4. Neither A nor B.

    Question 5
    Suppose A is an m × n matrix, where m < n. Which of the following statements is/are true?
    A. The nonhomogeneous system Ax = b has at least one solution.
    B. The homogeneous system Ax = 0 has infinitely many solutions
    1. Only A 2. Only B 3. Both A and B 4. Neither A nor B.
    I don't know this one, nor the rest below. please help!!!

    Question 6
    Suppose A is a 5 × 2 matrix, B is an r × s matrix and C is a 4 × 3 matrix. If A^T *B*C is defined, which of the following is true?
    1. the size of AT BC is 2 × 3
    2. r = 2, s = 5
    3. r = 3, s = 4
    4. r = 3, s = 5

    Question 7
    Suppose A is a 5×2 matrix, B is an r ×s matrix and C is a 4×3 matrix. If A^T *B*C is defined, then the size of B is
    1. 2 × 4
    2. 2 × 5
    3. 5 × 4
    4. 4 × 2

    Question 8
    Suppose A, B and C are invertible n × n matrices.
    Which of the following statements is/are true?
    A. (ABC)^T = A^T *B^T*C^T
    B. (ABC)^1 = A^1*B^1*C^1
    1. Only A 2. Only B 3. Both A and B 4. Neither A nor B.

    Question 9
    Which of the following statements is/are true?
    A.
    0 0 1
    0 1 0
    ?1 0 0
    is an elementary matrix.
    B. If E1 and E2 are two n × n elementary matrices then E1 × E2 is an elementary matrix.
    1. Only A 2. Only B 3. Both A and B 4. Neither A nor B.

    Question 10
    Suppose A is an upper triangular matrix. Which of the following is/are true?
    A. A has 0 as entry in each position below the main diagonal.
    B. A has 1 as entry in each position above the main diagonal.
    1. Only A 2. Only B 3. Both A and B 4. Neither A nor B.

    Thanks in advance!

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