MATH 3A, SSII 2024, Midterm Practice

Hello, dear friend, you can consult us at any time if you have any questions, add WeChat: daixieit

MATH 3A, SSII 2024, Midterm Practice

1. (24 points) True or False?

(1) any system of n linear equations in n variables has at most n solutions.

(2) If A is a 2 × 3 matrix then the system of equations Ax = 0 must have infinitely many solutions.

(3) If matrices A and B are row equivalent, they have the same reduced echelon form.

(4) if an n×n matrix A has n pivot positions, then the reduced echelon for of A is the n×n identity matrix.

(5) if a system of linear equations has two different solutions, it must have infinitely many solutions.

(6) If a system Ax = b has more than one solution, then so does the system Ax = 0.

2. (24 points) For the system x−y + 3z = 1, y = −2x+ 5, 3z −x−5y + 7 = 0, do the following

(1) write the system in the matrix form Ax = b, for x = .

(2) write out the augmented matrix for this system and calculate its reduced echelon form.

(3) write out the complete set of solutions (if they exist) in vector form using parameters if needed.

3. (18 points) Determine h and k such that the solution set of

x1 + 3x2 = k

4x1 + hx2 = 9

(a) is empty, (b) contains a unique solution, and (c) contains infinitely many solutions. (Hint: “empty” means “no solution”. Hint 2: For each of the cases (a), (b), or (c), you need to find all the possible h and k.)

4. (15 points) Let the matrix A be given by

Determine whether the columns of A span R4.

5. (19 points) The reduced row echelon form of a certain system of linear equations is:

Determine whether this system is consistent, and if so, find its general solution. In addition, write the solution in vector form.