Exercises: Rank of a matrix

In a digital marketing campaign, a company analyses the impact of two social media ads. The results are organised in a matrix $A = \begin{pmatrix} 120 & 45 \\ 80 & 30 \end{pmatrix}$, where the first column represents the "clicks" and the second the "purchases" made. The first row corresponds to ad 1 and the second to ad 2. The figure represents each ad as the vector (clicks, purchases) from the origin.

1. a) Calculate the determinant of the matrix $A$.

1. b) Explain whether there is any proportionality relationship between the data of the two ads and how it relates to the value of the determinant and to the figure.

Calculate the value of the following 3rd order determinant using Sarrus' rule:

State step-by-step the products of the main diagonals and the secondary diagonals.

Without performing complex calculations, justify using the properties of determinants why the value of the following determinant is zero:

A student is designing an algorithm for a video game and needs a transformation matrix to be "singular" (that is, its determinant is equal to zero). If the matrix is $M = \begin{pmatrix} x & 4 \\ 9 & x \end{pmatrix}$, find the possible real values of $x$ so that condition $|M| = 0$ is satisfied.

Determine the rank of the following matrix $B$ by studying its minors. What does the result obtained mean with regard to the linear independence of its rows?

Calculate the rank of the matrix $C$ using Gaussian elimination (creating zeros below the main diagonal):

Given the matrix $D$, which represents the stock of three types of sweatshirts in two different shops:

Calculate its rank and reason whether the information from the second shop is independent of the first or if, on the contrary, there is a proportional relationship between both.

Find the value of the parameter $k$ so that the rank of the following matrix is equal to 2:

Justify your answer by analysing the determinant of the matrix.

Carlos needs to determine the rank of the matrix $A = \begin{pmatrix} 1 & 2 & 0 \\ 2 & 4 & 1 \\ 3 & 6 & 2 \end{pmatrix}$. To do so, he calculates the determinant formed by the first two rows and the first two columns:

From this single calculation, Carlos concludes on his exam: "Since this minor of order 2 is zero, the rows are dependent and the matrix cannot have rank 2 or 3; therefore, its rank is directly 1".

1. a) Identify the conceptual error made by Carlos and explain why the fact that a specific minor vanishes does not guarantee that the rank is less than the order of that minor.
2. b) Find a non-zero minor of order 2 in matrix $A$ that disproves Carlos's conclusion and prove what the true rank of $A$ is using determinants.
3. c) What condition should all minors of order 2 have satisfied for Carlos's conclusion to be mathematically valid?

Consider the coefficient matrix $C \in \mathcal{M}_{3 \times 4}(\mathbb{R})$ associated with the production of three industrial plants:

1. a) Estimate: Before calculating any determinant, state between which theoretical minimum and maximum values $\operatorname{rg}(C)$ must lie, knowing that the matrix is not entirely zero. Look closely at the three rows: do you notice any obvious linear combination relationship between them at first glance? Based on that observation, make a reasoned estimate of the exact value of $\operatorname{rg}(C)$.
2. b) Check: Select a non-zero minor of order 2 to ensure that $\operatorname{rg}(C) \ge 2$. Then, border this minor with the remaining row and columns to calculate only the strictly necessary determinants of order 3. Determine the definitive rank of $C$ and assess whether your initial estimate was correct.

Given that the determinant of a square matrix $A$ of order 3 is $|A| = -2$, use the properties of determinants to calculate, giving reasons, the value of:

1. a) $|A^t|$ (determinant of the transpose).
2. b) $|A^{-1}|$ (determinant of the inverse).
3. c) $|3A|$.
4. d) The determinant of the matrix resulting from swapping the first two columns of $A$ and multiplying the third row by $-5$.

Determine the rank of the following matrix $M$ depending on the value of the real parameter $k$:

State for which values of $k$ the rank is maximum and what happens in the critical cases.

In a logistical transport network, the connections between three distribution centres ($A$, $B$ and $C$) can be modelled using an adjacency matrix. The following graph shows the active direct routes between the centres:

If the importance of each centre depends on the independence of its routes, given the route matrix $R$:

Calculate its determinant and determine the rank of matrix $R$. Pedagogically, what does it mean for the rank to be maximum in this connectivity context?

Find the values of $x$ for which the determinant of matrix $A$ is equal to zero:

Once found, explain what relationship exists between the rows of the matrix when $x = 1$.

Given the non-square matrix $B$ of dimensions $3 \times 4$:

1. a) Explain, giving reasons, why the rank of $B$ cannot be 4.
2. b) Calculate the rank of the matrix using the method of minors (bordering minors).

Consider matrices $A$ and $B$ such that:

1. a) Calculate $|A|$ and $|B|$.
2. b) Calculate $|A \cdot B|$ without performing the matrix multiplication, justifying which property you use.
3. c) Is the matrix $A$ invertible? What about the matrix $B$? Justify your answer based on the value of their determinants.

Three vectors in space $\mathbb{R}^3$ are given by the rows of the matrix $V$:

Determine for which values of the parameter $m$ these three vectors are linearly dependent. To do so, use the concept of determinant and its relationship with rank.

A video game company analyses its market share in three regions using a $S$ matrix. For the analysis system to be stable, the matrix must have rank 3.

1. a) Calculate the determinant of $S$ in terms of $m$.
2. b) Determine if there is any value of $m$ that would require the company to redesign its model due to having a rank less than 3. If so, what would the rank be in that case?

Consider the square matrix $M$, whose entries depend on two real parameters $a$ and $b$:

Answer the following questions with reasoned explanations:

1. a) Is it possible to find values for $a$ and $b$ such that the rank of $M$ is exactly 1? If possible, calculate them; if not, rigorously prove why by calculating a suitable minor.
2. b) Obtain the algebraic relationship that $a$ and $b$ must satisfy for the matrix to be singular (that is, for $\operatorname{rg}(M) < 3$).
3. c) Provide a specific pair of values $(a, b)$ that make $\operatorname{rg}(M) = 2$. For that pair of values, explicitly state a non-zero minor of order 2 and justify why it is not necessary to compute further determinants to ensure that the rank is 2.

A Spanish logistics company coordinates parcel shipping between three distribution centers located in Madrid, Zaragoza, and Barcelona. The daily allocation of vehicles is gathered in matrix $T$, whose rows represent the centers (Madrid, Zaragoza, and Barcelona, respectively) and whose columns represent the number of vehicles assigned to three types of routes (urban, regional, and long-distance):

where $k \in \mathbb{R}$ is a batch of vehicles on a regional route assigned to Barcelona, pending a technical audit. Management requires the three centers to have linearly independent operational profiles (rank 3) to ensure there are no redundant infrastructures.

1. a) Set up the determinant of $T$ in terms of $k$ and determine the critical value that would cause operational redundancy in the network (that is, such that $\operatorname{rg}(T) < 3$).
2. b) If the audit concludes that $k = 30$, calculate the rank of $T$ using bordered minors, interpret what this result means for the company's logistics, and make explicit the linear combination relationship linking Barcelona's operations with those of Madrid and Zaragoza.
3. c) If it is ultimately decided to set $k = 35$, is management's requirement of independence met? Justify your answer by calculating the value of the determinant.

Given the matrix $A$ of order $4 \times 4$ depending on a real parameter $a$:

Work out:

1. Calculate the determinant of $A$ using Gaussian elimination or by expanding along the elements of a row/column after creating zeros.

2. Determine the rank of matrix $A$ for the different values of $a$. Specify in which cases the matrix does not have an inverse.

3. For the value $a = 0$, is it possible to find a linear combination of the columns that yields the zero column? Justify your answer based on the value of the rank.

A cybersecurity company uses matrices to validate access keys. For a key to be secure, the rank of its associated matrix must be maximum. Consider the key matrix $K$:

1. Calculate the values of $k$ for which the key is not secure (that is, the rank of $K$ is less than 3).

2. If $k = -1$, study the linear dependence of the rows of the matrix. What is the rank in this case?

3. For the values of $k$ such that $|K| = 0$, state whether there exists any minor of order 2 that is non-zero.

Without expanding the determinant directly, use the properties of determinants to prove the following identity:

Subsequently, use this result to find the rank of matrix $M$ depending on the values of $x, y, z$:

In a social network study, the influence between three communities $C_1, C_2$ and $C_3$ is analysed. The interaction matrix $B$ is given by:

1. Determine for which values of $m$ the network is "redundant", that is, one of the communities can explain its behaviour as a linear combination of the other two.

2. For $m = -2$, calculate the rank of $B$ and find the linear dependence relationship between its rows.

3. If $|B| \neq 0$, what can we state about the set of vectors formed by the columns of $B$?

Given the matrix $A = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 4 \\ 5 & 6 & 0 \end{pmatrix}$, calculate $|A|$ and, using the properties of determinants, find $|A^T|$, $|A^{-1}|$, and $|A^2|$.

A matrix $Q$ is said to be idempotent if it satisfies $Q^2 = Q$.

1. Prove that the determinant of any idempotent matrix can only be $0$ or $1$.

2. If an idempotent matrix $Q$ of order 3 has determinant $|Q| = 1$, what must its rank be? Justify your answer.

3. Given the matrix $M = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ a & b & 0 \end{pmatrix}$, check whether it is idempotent and calculate its rank depending on $a$ and $b$.

Three friends, Alex, Bea and Iratxe, set up a food stall at their local neighbourhood festival and buy three types of ingredients ($X, Y, Z$). Matrix $V$ represents the units of each ingredient bought by each friend:

Where $a$ is the quantity of ingredient $Z$ bought by Bea.

1. Determine the value of $a$ for which Bea's shopping list is a linear combination of Alex's and Iratxe's lists. What happens to the rank of $V$ at that point?

2. If $a = 1$, calculate the inverse matrix $V^{-1}$.

3. For $a = -1$, justify whether the row vectors are linearly independent using the concept of determinant.

To calculate the rank of the rectangular matrix $P \in \mathcal{M}_{3 \times 4}(\mathbb{R})$:

two methodological procedures are compared:

• Strategy 1 (Exhaustive search): Calculate all $\binom{4}{3} = 4$ possible minors of order 3 formed by the columns of $P$.

• Strategy 2 (Bordered minors): Fix a non-zero minor of order 2 and systematically border it.

1. a) Reasonably compare both strategies in terms of efficiency: how many determinants of order 3 is it necessary to calculate in the best and worst cases with each method? Why does the bordering method avoid unnecessary calculations compared to testing minors at random?
2. b) Carry out Strategy 2: spot at first glance a proportionality relationship between two columns of $P$, strategically select a minor of order 2 that avoids complex calculations, border it with the remaining columns, and deduce the exact value of $\operatorname{rg}(P)$.
3. c) Imagine that a fourth row defined as $F_4 = F_1 + F_2$ is added to matrix $P$, obtaining a square matrix $Q$ of order $4 \times 4$. Without calculating any additional determinants, reason what the value of $\operatorname{rg}(Q)$ will be and justify which property of determinants or linear independence supports your conclusion.

In a study on economic transfers between three productive sectors, the parametric matrix $A(t) \in \mathcal{M}_{3 \times 3}(\mathbb{R})$ is used:

where $t \in \mathbb{R}$ is an intersectoral exchange rate.

1. a) Calculate the determinant of $A(t)$ by completely factoring it as a product of simple linear factors, and determine all values of $t$ for which the matrix is singular (that is, $\operatorname{rg}(A(t)) < 3$).
2. b) For each of the critical values found in the previous section, study the rank of the matrix by indicating in each case a non-zero minor of maximal order.
3. c) Inverse problem: Devise a condition on the parameter $t$ so that the rank is reduced to its minimum possible value. What is this minimum value and for what value of $t$ is it reached? Justify why the rank can never be 0.
4. d) Consider the column vector $B = \begin{pmatrix} 1 \\ 1 \\ 1 \end{pmatrix}$ and form the augmented matrix $M = (A(t) \mid B)$ of dimensions $3 \times 4$. Without performing determinant operations or row reduction, deduce in a purely reasoned manner the value of $\operatorname{rg}(M)$ in the cases $t = 1$ and $t = 0$, based on the linear dependence relationship between the columns of $A(t)$ and column $B$.

Topic challenge. Where is the labor market heading in your autonomous community? You will analyze employment dynamics by modeling them as a matrix system with official data.

1. Data collection: Access the INE website (EPA) and collect the quarterly transition rates between three states: employed, unemployed, and inactive. Construct a square matrix $P$ of order 3, verifying that its rows satisfy the properties of probabilities (values in $[0, 1]$ whose sum is $1,0$).

2. Algebraic analysis: In your notebook, find the determinant of $P$ using Laplace expansion supported by order 2 determinants. Calculate the rank of the matrix using the Gaussian elimination method and deduce, applying the properties of determinants, whether $P$ has an inverse matrix.

3. Simulation and report: In a spreadsheet, calculate $P^{-1}$ to estimate the population vector of the previous quarter from current data. Prepare a brief report (one page) that includes the calculations and interprets whether the labor system is stable or vulnerable.

Assessment criteria:

• Rigor in obtaining real data and formulating the stochastic matrix.

• Correct application of rank (Gauss), determinant (Laplace and order 2), and inverse.

• Critical socioeconomic interpretation of the results and spreadsheet proficiency.