Properties of Determinants

Knowing that the determinant of a square matrix $A$ of order 3 is $|A| = -4$, use the properties of determinants to calculate the value of the following determinants:

1. a) $|A^t|$, where $A^t$ is the transpose matrix of $A$.

1. b) The determinant of the matrix resulting from swapping the first two rows of $A$.

1. c) The determinant of the matrix obtained by multiplying the third column of $A$ by $-5$.

Knowing that $\begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = 6$, calculate the value of the following determinant without expanding it:

State step by step which properties you have used (swapping rows and multiplication by a scalar).

Without performing any operations or applying Sarrus' rule, justify why the following determinants are zero:

1. a) $\begin{vmatrix} 2 & 1 & 3 \\ 4 & 2 & 6 \\ 1 & -1 & 5 \end{vmatrix}$

1. b) $\begin{vmatrix} 1 & 5 & 0 \\ 2 & 3 & 0 \\ -4 & 7 & 0 \end{vmatrix}$

1. c) $\begin{vmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \\ 5 & 7 & 9 \end{vmatrix}$

Prove, using the properties of determinants, that if a linear combination of the other rows is added to a row of a square matrix, the value of the determinant does not change. To do so, check what happens to the determinant:

$|A| = \begin{vmatrix} a & b \\ c & d \end{vmatrix}$

if we perform the row operation $F_2 = F_2 + k \times F_1$.

Without expanding the determinant or using Sarrus' rule, show that the following determinant is zero for any value of the parameters $a, b, c \in \mathbb{R}$:

$\begin{vmatrix} 1 & a & b+c \\ 1 & b & a+c \\ 1 & c & a+b \end{vmatrix} = 0$

State at each step which elementary operation or property you are using.

Given the matrix $A = (a_{ij})$ of order 3, whose elements are given by the expression $a_{ij} = i + j$, that is,

Prove, using properties of lines (rows or columns) of determinants and without expanding it by Sarrus' rule, that its determinant is $|A| = 0$.

Let $M$ be a matrix of order $3 \times 3$ such that its determinant is $|M| = 2$. Calculate the value of the determinant of the matrix $3M$. Explain which property you have used and take care with the order of the matrix.

In a strategy video game, the coordinates of resources are stored in a matrix $3 \times 3$ called $C$. If we know that $|C| = 10$, what would be the value of the determinant if we double all the rows of the matrix (that is, we calculate $|2C|$)? Compare this result with the one obtained if we only doubled the first row.

An encryption system uses a matrix $A$ of order 3 to encode messages. If the determinant of the matrix is $|A| = k$, justify how the determinant changes if the following security modifications are made:

1. The first and third rows are swapped to increase entropy.

2. The second column is multiplied by $-1$ to change the polarity of the bits.

3. Three times the second row is added to the first row.

What would be the final value of the determinant in terms of $k$?

Given two square matrices $A$ and $B$ of the same order, we know that $|A| = 3$ and $|B| = -2$. Calculate, stating the properties applied:

1. a) $|A \times B|$

1. b) $|B^2|$

1. c) $|A \times B^t|$

If $A$ is a matrix of order 3 such that $|A| = \frac{1}{2}$, calculate the determinant of its inverse matrix $A^{-1}$, justifying your answer using the property of the determinant of the product of matrices.

Let $A$ be a square matrix of order 3 such that its determinant is $|A| = -2$. Using exclusively the properties of determinants, calculate the value of the following determinants:

1. a) $|A^t \times A|$

1. b) $|3 \times A^{-1}|$

1. c) $|\text{det}(A) \times A|$

Given the matrix $B$ of order 3, it is known that the determinant of its adjugate matrix is $|\text{Adj}(B)| = 64$. Calculate with reasoning the possible values of the determinant of the matrix $B$. Subsequently, calculate the determinant of the matrix $2B^t$.

In the design of a new video game, the position of an object in 3D space is defined by the determinant of a matrix $M$ representing its volume. It is known that:

$|M| = \begin{vmatrix} a & b & c \\ d & e & f \\ g & h & i \end{vmatrix} = 12$

Calculate, justifying the property applied, the value of the determinant of the matrix $N$ representing a deformation of the object:

$|N| = \begin{vmatrix} d & e & f \\ 2a-g & 2b-h & 2c-i \\ g & h & i \end{vmatrix}$

Solve the following equation for $x \in \mathbb{R}$ applying the properties of determinants to simplify the expression before calculating its value:

$\begin{vmatrix} x+1 & x & x \\ x & x+1 & x \\ x & x & x+1 \end{vmatrix} = 0$

If $A$ and $B$ are two square matrices of order 2 such that $|A| = 3$ and $|B| = -1$, calculate the value of the determinant of the matrix $X$ obtained from the following matrix equation:

$X = (A \times B)^t \times (2A)^{-1}$

Apply the properties of transposition, inversion and multiplication to simplify the expression before substituting the numerical values.

Let $A$ be a square matrix of order $n=4$ whose determinant is $|A| = \frac{1}{2}$. Using exclusively the properties of determinants, calculate with reasoning the value of the determinant of the matrix $B$, defined as:

$B = (2A)^{-1} \times A^t \times Adj(A)$

Where $A^t$ is the transpose of $A$ and $Adj(A)$ is its adjugate matrix.

Given two matrices $A$ and $B$ of order 3 such that $|A| = 3$ and $|B| = -2$, calculate the value of the determinant of the matrix $X$ satisfying the following matrix equation:

$A \times X \times B^{-1} = (A^t \times Adj(A))^{-1}$

Justify each property used in the process (determinant of a product, of an inverse, of a transpose, and the relationship between a matrix and its adjugate).

Prove, without expanding the determinant using Sarrus' rule or along a row/column, that the following determinant is zero for any values of the parameters $x, y, z$ such that $x+y+z=0$:

Hint: Try performing elementary operations between columns and rows to obtain common factors or proportional lines.

A computer vision algorithm for gesture recognition uses a transformation matrix $T$ of order 3. It is known that this matrix is skew-symmetric ($T^t = -T$).

1. a) Show with reasoning, using the properties of determinants, that the determinant of any skew-symmetric matrix of odd order is always 0.

1. b) If the matrix is modified to be $M = I + T$ (where $I$ is the identity matrix), what could you state about the invertibility of $M$ given that all the eigenvalues of $T$ are purely imaginary?

(Note for the student: Focus on part a) as the core of the determinant problem).

Solve the following equation for $x \in \mathbb{R}$ without fully expanding the polynomial, using the properties of rows/columns to simplify:

Hint: Observe the sum of the elements in each row or column.

Let $A$ be a matrix of order 3 whose elements are $a_{ij} = i + j$. Let $B$ be another matrix of order 3 whose elements are $b_{ij} = (-1)^{i+j}$.

1. a) Calculate $|A|$ and $|B|$ by applying linear combination properties of rows or columns.

1. b) Calculate the determinant of the matrix $C = 2A^2 \times B^t$.

1. c) Is it possible to find a matrix $D$ such that $A \times D = I$? Reason your answer based on the value of the determinant of $A$.

In the design of audio signal filters, a transfer matrix $M$ is used that depends on an angular frequency $\omega$. The matrix is given by:

$M(\omega) = \begin{pmatrix} 1 & \cos(\omega) & \cos(2\omega) \\ \cos(\omega) & \cos(2\omega) & \cos(3\omega) \\ \cos(2\omega) & \cos(3\omega) & \cos(4\omega) \end{pmatrix}$

Calculate the determinant of $M(\omega)$, simplifying the result as much as possible using trigonometric identities and properties of determinants.

Consider the matrix $A$ of order $n \times n$:

$A = \begin{pmatrix} 1+x & 1 & 1 & \dots & 1 \\ 1 & 1+x & 1 & \dots & 1 \\ 1 & 1 & 1+x & \dots & 1 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 1 & 1 & 1 & \dots & 1+x \end{pmatrix}$

Find the general expression for its determinant $|A|$ in terms of $n$ and $x$. Apply the result to find the determinant of a matrix of order 10 where $x=2$.