Calculating Determinants

A streamer is analyzing the growth of their channel. To compare the impact of two of their videos, they organize the "Likes" and "Shares" data into a matrix $A$ of order $2 \times 2$. Calculate the determinant of matrix $A$ to obtain an impact index:

Calculate the value of the determinant of matrix $B$ using Sarrus's rule. This matrix represents the inventory of components (chips, screens, and batteries) of three warehouses of an electronics 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 $(1, 5)$.

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

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

Given 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 row or column you consider most appropriate (Laplace expansion):

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

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

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

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 coordinate 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 fully simplified polynomial in $a$.

Without performing complex calculations, justify why the following determinants are equal to $0$ 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.

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:

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 $-5$.

Let matrix $A = (a_{ij})$ of order $3 \times 3$ be defined by $a_{ij} = i + j$. Also consider 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 matrix $M$, where:

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

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

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

In the development of a strategy video game, the position of three units on the map is 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 (aligned or linearly dependent units) in the following system:

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

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

1. a) Prove, without expanding the determinant using Sarrus's rule, that $W = (b-a)(c-a)(c-b)$.
2. b) If the nodes $a, b, c$ represent transmission frequencies, what condition must they meet 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 factor the resulting polynomial:

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

In the design of an aeronautical component, the 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 using Sarrus's 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?

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

1. a) Find the values of $\lambda$ for which 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 matrix $X$.

Analyze the rank of the following matrix $A$ in terms of 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 previously calculating the elements of the adjugate matrix. (Hint: use the relationship between a matrix, its adjugate, and its determinant).