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C1_W2_Practice-Quiz.md

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1. Solve the system of equations using the method of elimination and select the correct answer.

$$\begin{cases} x + y = 4 \cr -6x + 2y = 16 \end{cases}$$

  • The system has infinitely many solutions.
  • $x = 1, y = 3$
  • $x = -1, y = 5$
  • The system has no solution.
  • $x = 0, y = 0$

2. For the questions 2-3, calculate the determinant of the matrices and determine if the matrices are singular or non-singular:

$$\begin{bmatrix} 4 & -3 \cr 7 & -8 \end{bmatrix}$$

  • -53, Singular
  • -11, Non-singular
  • -53, Non-singular
  • -11, Singular

3.

$$\begin{bmatrix} -3 & 8 & 1 \cr 2 & 2 & -1 \cr -5 & 6 & 2 \end{bmatrix}$$

  • 36, Non-singular
  • -80, Non-singular
  • -20, Non-singular
  • 0, Non-singular
  • 0, Singular

4. Determine if the provided matrix has linearly dependent or independent rows (a, b, c, d, e, f are any real numbers):

$$\begin{bmatrix} a & b & c \cr d & e & f \cr 2a - d & 2b - e & 2c - f \end{bmatrix}$$

Hint: Can one row in the matrix be obtained as a result of operations on the other rows?

  • Independent
  • It cannot be determined.
  • Dependent

5. Which of the following operations, when applied to the rows of the matrix, do not change the singularity (or non-singularity) of the matrix:

  • Multiplying a row by a nonzero scalar.
  • Switching rows.
  • Adding a row to another one.
  • Adding a nonzero fixed value to every entry of the row.

6. In the following matrix:

$$\begin{bmatrix} a & a \cr b & c \end{bmatrix}$$

a, b, and c are non-zero real numbers. If the matrix is non-singular, which of the following must be true:

  • a = b only if c ≠ a
  • c = a only if a = b
  • c ≠ b
  • c = b

7. Luis went yesterday to the bank to find out the interest rate of three different financial instruments. He received the following information:

Financial instrument Savings account Certificate of Deposit (CD) Bonds
Annual interest 2% 3% 4%

He wants to invest his USD $10,000 savings in these three accounts. By doing so, he knows that after a year he would receive a total of US $ 260 in interest if he put twice as much money in the savings account as in the CDs, and “z” money in bonds.

Calculate the value of “z” , in USD, using the elimination method explained in the lectures.

  • $z = \text{USD}$ $5600
  • It cannot be determined.
  • $z = \text{USD}$ $1600
  • $z = \text{USD}$ $2800