Matrix Operations

In developing a new drone engine, an engineering team uses a transfer matrix $M$ that relates vibrations at three structural points. The matrix is given by:

$$M = \begin{pmatrix} 0 & a & 0 \\ 0 & 0 & b \\ 0 & 0 & 0 \end{pmatrix}$$

where $a$ and $b$ are non-zero real parameters.

1. a) Prove that matrix $M$ is nilpotent and determine its index of nilpotency (the smallest exponent $k$ such that $M^k = O$).
2. b) Use the previous result to calculate matrix $S = I + M + M^2 + M^3 + \dots + M^{100}$ in a simplified way, where $I$ is the identity matrix of order 3.

Given the dimensions of the following matrices:

• $A$ is a matrix of order $2 \times 3$

• $B$ is a matrix of order $3 \times 4$

• $C$ is a matrix of order $4 \times 2$

State, giving reasons, which of the following products can be performed and, if so, specify the dimension of the resulting matrix:

1. a) $A \times B$
2. b) $B \times A$
3. c) $(A \times B) \times C$

An online IT accessories store registers a customer's order: 3 wireless mice at $\text{RD\$}\ 720$ each, 5 USB-C cables at $\text{RD\$}\ 360$ each and 2 laptop stands at $\text{RD\$}\ 1\,500$ each.

Represent the quantities as a row matrix $Q = (3 \quad 5 \quad 2)$ and the unit prices as a column matrix $P$. Calculate the product $Q \times P$ to obtain the total order amount.

Calculate the product of matrices $A$ and $B$ in the order $A \times B$, where:

$A = \begin{pmatrix} 2 & -1 \\ 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 4 & 1 \\ -2 & 5 \end{pmatrix}$

A streaming platform analyses the consumption of two users over the weekend. Matrix $C$ represents the hours of series and films watched on Saturday (row 1) and Sunday (row 2):

$C = \begin{pmatrix} 3 & 2 \\ 1 & 4 \end{pmatrix}$

If the server maintenance cost per hour for series is $30$ pesos and for films is $48$ pesos, represented by the column matrix $P = \begin{pmatrix} 30 \\ 48 \end{pmatrix}$, calculate the product $C \times P$ and interpret the meaning of the elements of the resulting matrix.

Determine the value of the real number $x$ such that the following matrix equality holds:

$\begin{pmatrix} 1 & x \\ 2 & 0 \end{pmatrix} \times \begin{pmatrix} 3 \\ -1 \end{pmatrix} = \begin{pmatrix} 5 \\ 6 \end{pmatrix}$

In an electronic sports (eSports) competition, matrix $A$ records the scores of two teams (rows) in three different games (columns). Matrix $B$ indicates, for each game (rows), the bonus points of two types (columns): the first column is the points for victory and the second is the points for consistency.

Given the matrices:

$A = \begin{pmatrix} 2 & 1 & 3 \\ 0 & 2 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 2 \\ 2 & 0 \\ 3 & 1 \end{pmatrix}$

1. a) Calculate the product $C = A \times B$.

1. b) Interpret what the element $c_{11}$ of $C$ represents, taking into account that the rows of $A$ are the teams, the columns of $A$ and the rows of $B$ are the games, and the columns of $B$ are the types of bonus (points for victory and points for consistency).

Given matrix $M = \begin{pmatrix} 5 & 12 \\ -3 & 7 \end{pmatrix}$, verify that the identity element property holds by carrying out the operations:

1. a) $M \times I_2$
2. b) $I_2 \times M$

Where $I_2$ is the identity matrix of order 2.

Given the matrices $A = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 \\ 1 & -1 \end{pmatrix}$, show that matrix multiplication is not commutative by calculating:

1. a) $A \times B$
2. b) $B \times A$

Compare both results.

Determine the values of parameters $a$ and $b$ so that the following matrices $M$ and $N$ commute, that is, so that the equality $M \times N = N \times M$ holds:

$M = \begin{pmatrix} 1 & a \\ 0 & 1 \end{pmatrix}$ , $N = \begin{pmatrix} 2 & 3 \\ 0 & b \end{pmatrix}$

Given the matrices $A = \begin{pmatrix} 1 & 2 \\ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} -1 & 0 \\ 3 & 1 \end{pmatrix}$:

1. a) Calculate $(A+B)^2$.
2. b) Calculate $A^2 + 2AB + B^2$.
3. c) Compare both results and justify why, in matrix algebra, the algebraic identity for squaring a binomial does not hold in general. What condition would need to be met for it to be true?

Let the matrix be $A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$.

1. a) Calculate the powers $A^2$ and $A^3$.

1. b) Given the result, could you deduce the expression of $A^n$ for any natural number $n$?

The commuter train network of a city connects three stations ($A$, $B$ and $C$). The adjacency matrix $G$ represents direct links between the stations (where $g_{ij} = 1$ if there is a direct link from station $i$ to station $j$):

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

Calculate $G^2$ and $G^3$. Based on the meaning of the product of adjacency matrices, explain how many different ways there are to travel from station $A$ to station $B$ making exactly two changes (that is, in 3 journeys).

Given the square matrix $A = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}$:

1. a) Calculate $A^2$, $A^3$ and $A^4$.
2. b) Based on the previous results, conjecture a general formula for $A^n$ where $n \in \mathbb{N}$.
3. c) Prove the formula obtained using mathematical induction.

Let matrix $M = \begin{pmatrix} 2 & 1 \\ -3 & -2 \end{pmatrix}$ be given.

1. a) Calculate $M^2$. What is this type of matrix called?
2. b) Use the previous result to find, giving reasons and without carrying out long multiplications, the value of the power $M^{2025}$.

Given the square matrix of order 3:

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

Calculate the powers $A^2$, $A^3$ and $A^4$. From the results obtained, conjecture a general formula for the expression of $A^n$ in terms of $n \in \mathbb{N}$ and prove it using mathematical induction.

A social network analyzes the spread of "viral" content among four groups of users ($G_1, G_2, G_3, G_4$). The probability that a user in group $i$ shares the content with group $j$ in one hour is given by the weighted adjacency matrix $A$:

If the transition matrix is $A = \begin{pmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 1 & 0 & 0 & 0 \end{pmatrix}$:

1. a) Calculate $A^2$, $A^3$, and $A^4$. Interpret what the entry $a_{11}$ of the matrix $A^4$ means in the context of information dissemination.
2. b) Determine the matrix $B = A + A^2 + A^3 + A^4$. What does this matrix represent regarding the connections between the groups?

Consider the set of matrices of the form $A(x) = \begin{pmatrix} \cos x & -\sin x \\ \sin x & \cos x \end{pmatrix}$, representing rotations through an angle $x$ in the plane.

1. a) Prove that the product of two matrices of this type satisfies the property $A(x) \times A(y) = A(x+y)$.
2. b) Based on the previous part, find the expression for $[A(x)]^n$ for any $n \in \mathbb{N}$.
3. c) Determine the values of $x \in [0, 2\pi)$ for which $[A(x)]^3 = I$.

Given the matrices $A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}$ and $B = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \end{pmatrix}$:

1. a) Calculate the products $AB$ and $BA$. Do matrices $A$ and $B$ commute?
2. b) Calculate $(AB)^2$ and $A^2B^2$. Justify why the results are different using the properties of matrix multiplication.
3. c) Find the trace of matrix $C = B^{30}$.

An Amazon logistics center uses a matrix $P$ to organize the flow of packages between three loading bays. The flow matrix is $P = \begin{pmatrix} 0 & 1/2 & 1/2 \\ 1/4 & 0 & 3/4 \\ 1/2 & 1/2 & 0 \end{pmatrix}$.

A matrix is said to be stochastic if the sum of the elements in each of its rows is equal to 1.

1. a) Check that $P$ is a stochastic matrix.
2. b) Prove that the product of two stochastic matrices of order 3 is also a stochastic matrix.
3. c) Calculate $P^2$ and verify whether it maintains this property.

Let the matrix be $A = \begin{pmatrix} k & 0 \\ 1 & k \end{pmatrix}$ with $k \in \mathbb{R}$.

1. a) Calculate $A^2$, $A^3$ and $A^4$.

1. b) Determine the general expression for $A^n$.

1. c) In a quantum computing system, an operation matrix is required to have the sum of all its elements equal to 81 after 3 iterations (that is, in $A^3$). If $k > 0$, determine the value of the parameter $k$ for this to happen.

Find matrix $X$ satisfying the following matrix polynomial equation:

$X = A^2 - 5A + 2I$

where $A = \begin{pmatrix} 3 & -1 \\ 4 & 2 \end{pmatrix}$ and $I$ is the identity matrix of order 2.

If $A$ is a square matrix of order 3 such that it satisfies the relationship $A^2 - A + I = O$, where $I$ is the identity matrix and $O$ is the zero matrix:

1. a) Express $A^3$ as a linear combination of $A$ and $I$.
2. b) Express $A^4$ in terms of $A$ and $I$.
3. c) If $A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -1 & 1 & 1 \end{pmatrix}$, check whether the initial condition is satisfied.

Let $M$ be a square matrix of order 2 that satisfies the polynomial equation $M^2 - 5M + 6I = O$, where $I$ is the identity matrix and $O$ is the zero matrix.

1. a) If $M = \begin{pmatrix} 4 & x \\ 2 & y \end{pmatrix}$, find the values of $x$ and $y$ so that the equation is satisfied.
2. b) Using only the relation $M^2 = 5M - 6I$, express matrix $M^4$ as a linear combination of $M$ and $I$ (that is, find the scalars $\alpha$ and $\beta$ such that $M^4 = \alpha M + \beta I$).