Calculating 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 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 $(1, 5)$.

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 $0$:

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

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:

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:

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):

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

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

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.

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

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.

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 $C$:

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) $|3 \times A^t|$

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

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

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:

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.

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):

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.

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?

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.

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.

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 $A \times X \times A^t = I$, where $I$ is the identity matrix of order 3. Calculate $|X|$ indirectly before finding the matrix $X$.

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).