Exercises: Calculation of determinants

Un joven streamer está analizando el crecimiento de su canal. Para comparar el impacto de dos de sus vídeos, organiza los datos de "Likes" y "Compartidos" en una matriz $A$ de orden $2 \times 2$: cada fila corresponde a un vídeo y cada columna a un tipo de dato. Calcula el determinante de la matriz $A$ para obtener un índice de repercusión:

Calculate the value of the determinant of the matrix $B$ using Sarrus' rule. This matrix represents the inventory of components (chips, screens and batteries) of three warehouses of a tech store:

In a strategy video game, the power of a magic attack is calculated by solving an equation based on a determinant. Find the value of $x$ so that the following determinant is equal to $0$:

Without performing complex calculations, justify why the following determinants equal $0$ using the properties of determinants:

1. a) $D_1 = \begin{vmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 5 & 1 & 4 \end{vmatrix}$

1. b) $D_2 = \begin{vmatrix} 2 & -1 & 4 \\ 2 & -1 & 4 \\ 10 & 20 & 30 \end{vmatrix}$

1. c) $D_3 = \begin{vmatrix} 1 & 3 & 2 \\ 2 & 6 & 4 \\ 0 & 1 & 5 \end{vmatrix}$

A logistics company uses an upper triangular matrix to organize its delivery routes efficiently. Calculate the determinant of the matrix $L$, knowing that for this type of matrix the calculation is greatly simplified:

Given the matrix $M$, calculate its minor $\alpha_{21}$ and its cofactor $A_{21}$:

We know that the determinant of a matrix $A$ of order $2 \times 2$ is $|A| = 5$. If we multiply all the elements of the second row by $3$, what will be the value of the new determinant? Justify your answer based on the properties.

In a graphic design program, the area of a triangle with one vertex at the origin $(0,0)$ and the other two at $(x_1, y_1)$ and $(x_2, y_2)$ can be calculated as half the absolute value of the determinant formed by their coordinates. Calculate the area of the triangle formed by the points $(0,0)$, $(4, 2)$ and $(1, 5)$.

Mateo needs to calculate the determinant of the matrix $A = \begin{pmatrix} 3 & -2 & 5 \\ 0 & 4 & 0 \\ 1 & -1 & 2 \end{pmatrix}$. To save time, he notices that the second row contains two zeros and decides to expand using Laplace's formula along that row. His written expansion is:

Convinced of his method, in the next exercise he applies the exact same rule to compute the determinant of the matrix $B = \begin{pmatrix} 1 & 0 & 4 \\ 2 & 3 & -1 \\ 0 & 5 & 0 \end{pmatrix}$, expanding along the third row:

When correcting his work, his teacher points out that the value obtained for $|A|$ was coincidentally correct due to the position of the element, but that he made an important conceptual error in $|B|$.

a) Identify the conceptual error made by Mateo when calculating $|B|$, explaining with the definition of cofactor ($A_{ij} = (-1)^{i+j} \cdot \alpha_{ij}$) which element he omitted.

b) Justify why his procedure did not alter the final numerical result for matrix $A$.

c) Correct the calculation of $|B|$ and state its actual value.

To calculate the determinant of the $4 \times 4$ matrix:

two different approaches are proposed:

• Strategy 1: Perform elementary row operations using Gaussian elimination until an upper triangular matrix is obtained, then multiply the elements on the main diagonal.

• Strategy 2: Apply Laplace expansion using the cofactors of the second row, reducing the calculation directly to a determinant of order 3.

a) Compare both strategies and explain which one is significantly faster and more efficient for this specific matrix based on the number of zeros it contains.

b) Following the strategy you selected as preferred, calculate the determinant of $M$ step by step, showing the intermediate calculations in detail.

Given the square matrix $A$ of order 3, it is known that its determinant is $|A| = -2$. Use the properties of determinants to calculate with reasoning the value of:

a) $|A^3|$

b) $|3 \cdot A^t|$

c) $|A^{-1} \cdot (2A)|$

d) The determinant of the matrix resulting from swapping the first two rows of $A$ and multiplying the third row by $-5$.

Find all real values of $x$ for which the following equality between determinants is satisfied:

Calculate the value of the determinant of the matrix $M$ of order 4 by first transforming it into an upper triangular matrix using Gaussian elimination (creating zeros below the main diagonal):

Knowing that the determinant of a square matrix $B$ of order 3 is:

Calculate, without expanding by Sarrus' rule, the value of the determinant of the matrix $C$:

Determine for which values of the real parameter $k$ the matrix $A$ does not have an inverse. To do this, set its determinant to zero and solve the resulting equation:

In the development of a strategy video game, the positions of three units on the map are given by the rows of a matrix. The determinant of this matrix indicates the battalion's area of influence. Calculate the value of $a$ so that the area of influence is zero (units collinear or linearly dependent) in the following system:

Note: This is a Vandermonde determinant. Try to simplify it using column properties before calculating it.

Calculate the value of the following determinant of order 4 by expanding along the elements of the most suitable row or column (Laplace expansion):

Solve the matrix equation $|X| = 0$, where $X = A - \lambda I$, $A$ being the given matrix and $I$ the identity matrix of order 3. Find the values of $\lambda$ that make the determinant zero (eigenvalues):

You want to design a $3 \times 3$ matrix $A = (a_{ij})$ with non-negative integer entries ($a_{ij} \in \{0, 1, 2, \dots\}$) intended for an encryption algorithm, subject to the following simultaneous conditions:

1. The matrix $A$ is upper triangular (that is, $a_{ij} = 0$ whenever $i > j$).

2. The sum of the three elements on its main diagonal is equal to $7$.

3. Its determinant is exactly $|A| = 12$.

4. The cofactor of entry $a_{11}$ is $A_{11} = 6$.

a) Deduce with justification the unique value of $a_{11}$ using the properties of the determinant of a triangular matrix and the associated minor.

b) Determine the possible values of the remaining diagonal elements $a_{22}$ and $a_{33}$.

c) Write a specific matrix $A$ that satisfies all the required conditions and verify explicitly, using the product of its diagonal and the calculation of $A_{11}$, that the results match the given data.

A technical team from the public company Tragsa is preparing a Cadastral report on an olive grove plot in the province of Jaén. On the official topographic map, the boundaries of the estate form a triangle whose three vertices have coordinates $A(1, 1)$, $B(7, 3)$, and $C(3, 8)$ (measured in hectometres, $\text{hm}$).

In cartography, the area of a planar triangular region with vertices $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$ is calculated using the formula:

a) To calculate $|M|$ efficiently without using Sarrus' rule, apply elementary row operations ($F_2 \to F_2 - F_1$ and $F_3 \to F_3 - F_1$) to create two zeros in the third column, and then expand using the cofactors of that column.

b) Calculate the resulting area of the plot and interpret the result by expressing it in hectares ($1\text{ hm}^2 = 1\text{ ha}$).

c) The adjacent owner claims that an irrigation shed located at the coordinates $D(5, 2)$ lies exactly on the straight boundary line connecting $A$ and $B$. Set up the corresponding determinant to check the collinearity and conclude with justification whether the statement is true or false without plotting the points.

Let $A = (a_{ij})$ be the matrix of order $3 \times 3$ defined by $a_{ij} = i + j$. Consider also a matrix $B$ such that its determinant is $|B| = 2$. Using exclusively the properties of determinants, calculate with reasoning the value of the determinant of the matrix $M$, where:

Explain whether the result obtained allows us to state that the matrix $M$ has an inverse.

In the design of an aeronautical component, structural stability is given by the determinant of a stress matrix $T$. It is known that the determinant of the following matrix depends on a load parameter $k$:

1. a) Calculate the value of the determinant in terms of $k$ using the properties of determinants to simplify the calculation (avoid direct expansion by Sarrus' rule).
2. b) Determine for which values of $k$ the component is unstable ($|T| = 0$).
3. c) If $k=3$, what relationship exists between the rows of the matrix?

A GPS positioning system uses four satellites to triangulate a position in space. The volume of the tetrahedron formed by the four satellites is related to the absolute value of the determinant of their coordinates matrix. Calculate with reasoning the value of the following determinant of order 4, which represents a specific configuration of these satellites:

Express the result as a polynomial in $a$ simplified as far as possible.

Given the matrix depending on a real parameter $\lambda$:

1. a) Find the values of $\lambda$ for which the matrix $A(\lambda)$ does not have an inverse.
2. b) For $\lambda = 0$, solve the matrix equation $A \cdot X \cdot A^t = I$, where $I$ is the identity matrix of order 3. Calculate $|X|$ indirectly before finding the matrix $X$.

Let $C_1, C_2, C_3$ be the three columns of a square matrix $M$ of order 3, such that its determinant is $|M| = -5$. Calculate, justifying each step with the corresponding property, the value of the determinant of the matrix whose columns are:

In a telecommunications network, flow capacity is modelled using the determinant of the adjacency matrix $W$. $W$ is defined as:

1. a) Show that, without expanding the determinant by Sarrus' rule, $W = (b-a)(c-a)(c-b)$.
2. b) If the nodes $a, b, c$ represent transmission frequencies, what condition must they satisfy for the determinant to be non-zero? Justify the importance of this condition for the independence of the signals.

Solve the following cubic equation using the properties of determinants to factorise the resulting polynomial:

State step by step the elementary operations performed on the rows or columns.

Determine the rank of the following matrix $A$ according to the values of the real parameters $m$ and $n$:

For the case where $m=1$ and $n=2$, calculate the determinant of the adjugate matrix of $A$, that is, $|Adj(A)|$, without calculating the elements of the adjugate matrix beforehand. (Hint: use the relationship between a matrix, its adjugate and its determinant).

Consider the $4 \times 4$ symmetric matrix:

a) Estimate: If all the off-diagonal elements were zeros, the resulting matrix would be diagonal and its determinant would be $4^4 = 256$. Reason conceptually whether you expect the actual determinant $|C|$ to be greater than or less than $256$, considering that the presence of positive off-diagonal values makes the rows "more dependent" on each other.

b) Calculate: To triangularize the matrix without introducing fractions:

1. Replace the first row with the sum of all four rows ($F_1 \to F_1 + F_2 + F_3 + F_4$) and factor out the resulting common factor from the determinant.

2. In the remaining matrix, subtract the first row from each of the other rows ($F_i \to F_i - F_1$ for $i = 2, 3, 4$) to obtain an upper triangular matrix.

3. Obtain the exact numerical value of $|C|$ by multiplying the diagonal elements.

c) Check and generalize: Compare the exact value with your initial estimate. Then, generalize the reasoning to derive a closed formula for the determinant of an $n \times n$ matrix that has the value $k$ along the entire main diagonal and the value $1$ in all remaining positions.

Consider the $4 \times 4$ matrix depending on the real parameter $k$:

Two students discuss the best strategy to compute $|A(k)|$ and analyse for which values of $k$ the matrix lacks an inverse:

• Álvaro proposes triangularizing using Gaussian elimination by dividing the first row by $k$ to eliminate the ones in the first column.

• Beatriz argues that dividing by $k$ is problematic if the case $k = 0$ is not analysed separately, and maintains that expanding by Laplace along rows or columns with zeros (or subtracting rows from each other) avoids any division by the parameter.

a) Justify why Álvaro's proposal is methodologically incomplete if it does not distinguish cases, explaining the algebraic risk of performing elementary operations that involve dividing by a parametric expression.

b) Following Beatriz's strategy, calculate $|A(k)|$ using Laplace expansion or row combinations that do not require division, detailing the cofactors calculated until obtaining the factorized polynomial in terms of $k$.

c) Determine all real values of $k$ for which the matrix $A(k)$ is not invertible, analysing what happens to the rank of the matrix when $k = 0$.