Exercises: 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 $-5$.

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

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.

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) $|A \cdot B|$

1. b) $|B^2|$

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

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.

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 $F_2 = F_2 + k \cdot F_1$.

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.

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).

Lucía is preparing for the university entrance exam and comes across the following problem: "Let $A$ be a square matrix of order 3 with $|A| = 4$. Calculate $|-2A|$ and $|A + A|$". In her notebook, she writes the following solution:

1) $|-2A| = -2 \cdot |A| = -2 \cdot 4 = -8$.

2) $|A + A| = |A| + |A| = 4 + 4 = 8$.

Identify, with reasoned arguments, the two conceptual mistakes made by Lucía, explain which formal properties of determinants she ignored or applied incorrectly, and write the rigorous mathematical correction to obtain the correct values of both determinants.

Given a square matrix $M$ of order 3 with known determinant $|M| = -2$, you want to transform it into a new matrix $P$ whose determinant is exactly $|P| = 12$.

Design a sequence of exactly three successive elementary transformations (row or column operations) that allow you to obtain $P$ from $M$. You must specify clearly what operation you perform at each step, justify how the determinant changes by applying the corresponding properties, and verify that the final result is indeed 12.

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) $|A^t \cdot A|$

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

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

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:

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

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.

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

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

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 $2B^t$.

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 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 $|A| = 0$.

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:

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

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

You want to calculate the fully factored expression of the determinant of the following matrix of order 3:

To approach it, two procedures are proposed:

• Method 1 (Sarrus' rule and subsequent algebraic factorization): Directly expand the six diagonal products and then group terms in the resulting polynomial to factor out common factors by trial and error.

• Method 2 (Row properties and cofactor expansion): Perform the elementary transformations $F_2 \to F_2 - F_1$ and $F_3 \to F_3 - F_1$, apply the properties of factoring out scalars from rows, and expand along the first column.

Critically compare both methods by analyzing their computational cost and the risk of algebraic error. Justify which of the two is more suitable and apply it step by step, stating each determinant property you use until you obtain the fully factored polynomial.

The National Statistics Institute (INE) studies annual labor transitions among three economic sectors in Spain (Agriculture, Industry, and Services) using a column-stochastic matrix $P$, of dimension $3 \times 3$, where each entry $p_{ij}$ represents the proportion of employed individuals who move from sector $j$ to sector $i$. Since the sum of the proportions in each column is always 1, in the matrix $P - I$ (where $I$ is the identity matrix of order 3) the sum of the elements in each column is equal to 0.

1. a) Using the properties of the determinant of the transpose matrix and the vanishing of determinants when a linear combination exists between rows or columns, formally prove that $|P - I| = 0$.
2. b) If the survey records yield $|P| = 0,25$, calculate, with reasoned steps, the determinant of the matrix $Q = 3 \cdot P^2 \cdot (P^t)^{-1}$, stating each property used, and interpret whether the transition matrix $P$ has an inverse matrix in this demographic model.

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:

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

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

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.

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.

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:

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

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).

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 $x=2$.

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 $C = 2A^2 \cdot B^t$.

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

Consider a square matrix $A$ of order $n = 4$ whose determinant is $|A| = -2$. From it, the matrix $B = -A^3 \cdot (2A^t)^{-1}$ is defined.

1. a) Estimate: Without carrying out detailed numerical operations, qualitatively anticipate:

1. Will the sign of $|B|$ be positive or negative? Justify your hypothesis considering the sign of $|A|$, the effect of the cubic power, the overall negative sign, and matrix inversion.

2. Will the absolute value of $|B|$ be strictly greater than, equal to, or less than 1? Support your conjecture by mentally comparing the magnitude of the powers of base 2 that will appear in the numerator and in the denominator.

1. b) Calculate: Obtain the exact numerical value of $|B|$, deriving each step from the properties of the determinant of a product, matrix power, scalar multiplication in order 4, transpose, and inverse.
2. c) Check and reflect: Contrast the result with your initial estimates and analyze which are the two most common conceptual errors usually made with the sign $(-1)$ and with the constant 2 in matrices of even dimension.

A square matrix $A$ of dimension $n$ is said to be skew-symmetric (or antisymmetric) if it equals the opposite of its transpose, that is, if it satisfies the matrix equality $A^t = -A$.

1. a) Relying rigorously on the properties $|A^t| = |A|$ and $|k \cdot M| = k^n \cdot |M|$, formally prove that the determinant of any skew-symmetric matrix of odd order $n$ is necessarily zero.
2. b) Inverse problem: The aim is to determine whether it is possible to find a skew-symmetric matrix $A$ of order $n \in \{2, 3, 4, 5\}$ whose determinant is exactly $|A| = 16$.

1. Deduce which dimensions from the given set are automatically ruled out and why.

2. For the smallest admissible order, set up the general algebraic structure with real coefficients of such a skew-symmetric matrix, compute its determinant symbolically, and find a concrete numerical example whose elements satisfy the condition $|A| = 16$.