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# Algebra (multiple choice) questions on matrices

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1. 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.

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