Q: Let $$A = \begin{bmatrix} 3 & 2 \\ 0 & 1 \\ 1 & -1 \end{bmatrix}$$ and $$B = \begin{bmatrix} 4 & 1 \\ 5 & -7 \end{bmatrix}. $$What is $$AB$$?

$$\begin{bmatrix} 22 & -11 \\ 5 & -7 \\ -1 & 8 \end{bmatrix}$$

Q: What is the inverse of $$B = \begin{bmatrix} 4 & 1 \\ 5 & -7 \end{bmatrix}$$?

$$\frac{1}{-33}\begin{bmatrix} -7 & -1 \\ -5 & 4 \end{bmatrix}$$

Question 1

Which of the following is the correct domain of the function $$f(x) = \log(3x - 4) in $$interval notation?

Accepted Answer

$$\left(\frac{4}{3}, \infty\right)$$

Question 2

Suppose you invest $5000 in an account earning 7% annual interest compounded daily. Which formula should you use to find the amount after 3 years?

Accepted Answer

$$A = P\left(1 + \frac{r}{n}\$$right)^{nt}$$

Question 3

If $$A = 5000\left(1 + \frac{0.07}{365}\$$right)^{365 	imes 3}$$, what is the value of A$ rounded to the nearest cent?

Accepted Answer

$$A \approx 6122.55$$

Question 4

Given the formula for APY (Annual Percentage Yield) for interest compounded $$n$$ times per year: $$APY = \left(1 + \frac{r}{n}\right)^n - 1$$, what is the APY for a 7% rate compounded daily (use $$n=365$$)?

Accepted Answer

$$APY \approx 0.0725$$

Question 5

How long will it take for $$5000 to grow to 8000 at 7$$% interest compounded daily? Use $$A = P\left(1 + \frac{r}{n}\$$right)^{nt}$$ and solve for t$.

Accepted Answer

$$t \approx 6.8$$ years

Question 6

Write the expression $$\frac{1}{2}\log x + 3\log y as a $$single logarithm.

Accepted Answer

$$\log\$$left(y^3$$ x$$^{1/2}$$\right)$$

Question 7

Find the inverse of the function $$f(x) = 4x + 7.$$

Accepted Answer

$$f^{-1}(x$$) = \frac{x - 7}{4}$$

Question 8

Find the inverse of $$g(x) = 4\log(x - 7).$$

Accepted Answer

$$g^{-1}(x$$) = $$10^{\frac{x}$${4}} + 7$$

Question 9

Solve for $$x$$: $$2(3^x) + 1 = 29.$$

Accepted Answer

$$x = 2$$

Question 10

Solve for $$x$$: $$4\ln(5x) = 12.$$

Accepted Answer

$$x = e$$^{3}$$/5$$

Question 11

Solve for $$x$$: $$\log_4$(x - 2) - \log_4$(x - 1) = 1.$$

Accepted Answer

$$x = 6$$

Question 12

Given the system $$\begin{cases} x^2$ - y^2$ = 2 \ x^2$ + y^2$ = 4 \end{cases}$$, what is one solution for $$(x, y)$$?

Accepted Answer

$$(\sqrt{3}, 1)$$

Question 13

Given the system of equations:
$$\begin{cases} 2x + 4y - 5z = 10 \ 2x + 4y - 2z = 8 \ x - y + 2z = -4 \end{cases}$$
What is the solution $$(x, y, z)$$?

Accepted Answer

$$(2, 0, -2)$$

Question 14

Let $$B = \begin{bmatrix} 4 & 1 \ 5 & -7 \end{bmatrix}. $$What is $$\det B$$?

Accepted Answer

$$-33$$

Question 15

Let $$A = \begin{bmatrix} 3 & 2 \ 0 & 1 \ 1 & -1 \end{bmatrix}$$ and $$B = \begin{bmatrix} 4 & 1 \ 5 & -7 \end{bmatrix}. $$What is $$AB$$?

Accepted Answer

$$\begin{bmatrix} 22 & -11 \ 5 & -7 \ -1 & 8 \end{bmatrix}$$

Question 16

What is the inverse of $$B = \begin{bmatrix} 4 & 1 \ 5 & -7 \end{bmatrix}$$?

Accepted Answer

$$\frac{1}{-33}\begin{bmatrix} -7 & -1 \ -5 & 4 \end{bmatrix}$$

Question 17

Find the partial fraction decomposition of $$\frac{7x - 1}{2x^2$ - 3x}.$$

Accepted Answer

$$\frac{2}{x} + \frac{3}{2x - 3}$$

Precalculus College Algebra Review: Exponential, Logarithmic, and Matrix Problems

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