Q: Let $$A = \begin{pmatrix} 1 & 4 \\ 2 & 7 \end{pmatrix}. $$What is $$A^{-1}$$?

$$\begin{pmatrix} -7 & 4 \\ 2 & -1 \end{pmatrix}$$

Q: Let $$A = \begin{pmatrix} 1 & 3 \\ 0 & 1 \end{pmatrix}. $$What is $$A^5$$?

$$\begin{pmatrix} 1 & 15 \\ 0 & 1 \end{pmatrix}$$

Question 1

Let $$\mathbf{u} = (3, 4)$$ and $$\mathbf{v} = (2, -7) in$$ $$\mathbb{R}^2. $$Which of the following is the correct expression for $$\mathbf{u} + \mathbf{v} as a $$linear combination of the standard basis vectors $$\mathbf{i}$$ and $$\mathbf{j}$$?

Accepted Answer

$$5\mathbf{i} - 3\mathbf{j}$$

Question 2

A weather station records wind as a vector $$\mathbf{w} = (-2, 1) in $$the plane. What is the magnitude of this wind vector?

Accepted Answer

$$\sqrt{(-2)^2$ + 1^2$} = \sqrt{4 + 1} = \sqrt{5}$$

Question 3

Let $$\mathbf{u} = (3, 5)$$ and $$\mathbf{v} = (-1, 2). $$Calculate $$\| -3\mathbf{u} + 2\mathbf{v} \|.$$

Accepted Answer

$$11\sqrt{2}$$

Question 4

Which of the following statements about the dot product in $$\mathbb{R}^2 is $$correct?

Accepted Answer

$$\mathbf{u} \cdot \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \cos 	heta$$

Question 5

Let $$z = 3 - 2i. $$What is the multiplicative inverse $$z^{-1} in $$standard form?

Accepted Answer

$$\frac{3}{13} + \frac{2}{13}i$$

Question 6

A complex number $$z = 1 + i is $$written in polar form. What is its modulus and principal argument?

Accepted Answer

Modulus $$\sqrt{2}$$, argument $$\frac{\pi}{4}$$

Question 7

Which of the following is the correct formula for the determinant of a $$2 	imes 2$$ matrix $$A = \begin{pmatrix} a & b \ c & d \end{pmatrix}$$?

Accepted Answer

$$ad - bc$$

Question 8

Given the system of equations:

$$\begin{cases} x + y + 2z = 8 \ -x - 2y + 3z = 1 \ 3x - 7y + 4z = 10 \end{cases}$$

After Gaussian elimination, what is the solution for $$x$$?

Accepted Answer

$$x = 3$$

Question 9

Let $$A = \begin{pmatrix} 1 & 4 \ 2 & 7 \end{pmatrix}. $$What is $$A^{-1}$$?

Accepted Answer

$$\begin{pmatrix} -7 & 4 \ 2 & -1 \end{pmatrix}$$

Question 10

Let $$\mathbf{a} = (2, -1, 1)$$ and $$\mathbf{b} = (1, 1, 2) in$$ $$\mathbb{R}^3. $$What is the angle $$	heta$$ between $$\mathbf{a}$$ and $$\mathbf{b}$$?

Accepted Answer

$$	heta = \frac{\pi}{3}$$

Question 11

Which of the following is the correct geometric interpretation of the cross product $$\mathbf{a} 	imes \mathbf{b} in$$ $$\mathbb{R}^3$$?

Accepted Answer

It is a vector perpendicular to both $$\mathbf{a}$$ and $$\mathbf{b}$$ whose magnitude equals the area of the parallelogram formed by $$\mathbf{a}$$ and $$\mathbf{b}.$$

Question 12

Let $$A = \begin{pmatrix} 2 & 0 & 0 \ 3 & 4 & 0 \ -1 & 1 & -2 \end{pmatrix}. $$What is $$\det(A)$$?

Accepted Answer

$$-16$$

Question 13

Suppose you have a system of linear equations $$AX = B$$ where $$A is a$$ $$3 	imes 3$$ matrix with $$\det(A) = 0. $$What can you conclude about the solutions?

Accepted Answer

The system has either no solution or infinitely many solutions.

Question 14

Let $$z = 2e$$^{i\frac{2\pi}$${3}}$$ and $$w = 2e$$^{i\frac{5\pi}$${6}}. $$What is $$\frac{z}{w} in $$Euler's form?

Accepted Answer

$$e^{-i\frac{\pi}$${6}}$$

Question 15

Which of the following is the correct formula for the projection of vector $$\mathbf{u}$$ onto vector $$\mathbf{v} in$$ $$\mathbb{R}^3$$?

Accepted Answer

$$\mathrm{proj}_{\mathbf{v}} \mathbf{u} = \frac{\mathbf{u} \cdot \mathbf{v}}{\|\mathbf{v}\|^2} \mathbf{v}$$

Question 16

Let $$A = \begin{pmatrix} 1 & 3 \ 0 & 1 \end{pmatrix}. $$What is $$A^5$$?

Accepted Answer

$$\begin{pmatrix} 1 & 15 \ 0 & 1 \end{pmatrix}$$

Question 17

A medical imaging system uses the vector $$\mathbf{v} = (6, 3, -2) to $$represent a direction in space. What is the unit vector in the direction of $$\mathbf{v}$$?

Accepted Answer

$$\frac{1}{7}(6, 3, -2)$$

Vectors, Complex Numbers, Matrices, and Systems of Equations: College Algebra Study Notes

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