Matrices and algebraic structures: introduction

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1. Question 1Let $A = \begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ 2 & 1 \end{pmatrix}$. What is the product $AB$?

   $\begin{pmatrix} 4 & 3 \\ 8 & 7 \end{pmatrix}$

   $\begin{pmatrix} 3 & 4 \\ 5 & 8 \end{pmatrix}$

   $\begin{pmatrix} 0 & 2 \\ 6 & 4 \end{pmatrix}$

   $\begin{pmatrix} 2 & 4 \\ 4 & 10 \end{pmatrix}$

2. Question 2Consider the system of equations $x + y + z = 1$, $x + 2y + 3z = 2$ and $2x + 3y + 4z = k$. For which values of $k$ is it consistent, and how many solutions does it then have?

   Only for $k = 3$, and then it has infinitely many solutions, which depend on one parameter.

   For every $k$, with a unique solution.

   Only for $k = 3$, and then it has a unique solution.

   For no $k$.

3. Question 3Compute the determinant of the matrix with rows $\left( 2, 1, 1 \right)$, $\left( 1, 3, -1 \right)$ and $\left( 0, 1, 4 \right)$.

4. Question 4Let $A$ be a matrix of order $3$ with $\det A = 2$. What is the value of $\det\left( 3 A^{\mathsf{T}} \right)$?

5. Question 5What is the inverse of the matrix $A = \begin{pmatrix} 3 & 5 \\ 1 & 2 \end{pmatrix}$?

   $\begin{pmatrix} 2 & -5 \\ -1 & 3 \end{pmatrix}$

   $\begin{pmatrix} 2 & -1 \\ -5 & 3 \end{pmatrix}$

   $\begin{pmatrix} \frac{1}{3} & \frac{1}{5} \\ 1 & \frac{1}{2} \end{pmatrix}$

   $\begin{pmatrix} 2 & 5 \\ 1 & 3 \end{pmatrix}$

6. Question 6Let $A$, $B$ and $C$ be square matrices of the same order, with $A$ and $B$ invertible. What is the solution of the equation $AXB = C$?

   $X = A^{-1} C B^{-1}$

   $X = B^{-1} C A^{-1}$

   $X = A^{-1} B^{-1} C$

   $X = \frac{C}{AB}$

7. Question 7What is the matrix of the transformation of the plane that rotates each vector a quarter turn counterclockwise about the origin and then reflects it in the horizontal axis?

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

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

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

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

8. Question 8Let $M(z)$ be the matrix of multiplication by the complex number $z$. Compute $\det M(3 - 4i)$.

9. Question 9Which of these sets, with the operation indicated, is a group?

   The nonzero remainders of $\mathbb{Z}_7$, with multiplication.

   The integers, with multiplication.

   The natural numbers, with addition.

   The nonzero remainders of $\mathbb{Z}_6$, with multiplication.

10. Question 10A city has $600$ thousand inhabitants in its center and $400$ thousand in its outskirts. Each year, $10$ percent of the inhabitants of the center move to the outskirts and $20$ percent of those of the outskirts move to the center; nobody else arrives or leaves. If the proportions remain the same, how many thousand inhabitants will the center have after two years?

11. Question 11From earlier unitsSolve the system $2x + 3y = 7$, $x - y = 1$ by Cramer’s rule and give the value of $x$.

12. Question 12From earlier unitsFind the inverse of $4$ modulo $9$, that is, the remainder $x$ between $0$ and $8$ such that $4x \equiv 1 \pmod{9}$.

13. Question 13From earlier unitsWhat is the polar form of $1 + i\sqrt{3}$?

    $2\left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)$

    $2\left( \cos \frac{\pi}{6} + i \sin \frac{\pi}{6} \right)$

    $4\left( \cos \frac{\pi}{3} + i \sin \frac{\pi}{3} \right)$

    $2\left( \cos \frac{2\pi}{3} + i \sin \frac{2\pi}{3} \right)$