Exercises: Properties of determinants

Suppose you are analysing a matrix $A$ of order $2 \times 2$ representing the profits of two branches of a clothing store in Madrid and Barcelona. The determinant of this matrix is $|A| = 12$.

Calculate, without performing the full expansion, the value of the following determinants by applying the corresponding properties:

1. a) The determinant of the transpose of the matrix: $|A^t|$.
2. b) The determinant of the matrix if we swap the Madrid row with the Barcelona row.
3. c) The determinant of the matrix if the profits for both cities were doubled (that is, calculating $|2 \cdot A|$).

A graphic design application stores the coordinates of a logo in a matrix $M$ of order 3. The system informs us that the determinant of this matrix is $|M| = 0$.

Explain with reasons what you can state about the rows of the matrix $M$. Does the inverse matrix $M^{-1}$ exist? Justify your answer based on the properties of determinants.

Given the following matrix $B$:

$B = \begin{pmatrix} 1 & 5 & -2 \\ 0 & 3 & 4 \\ 1 & 5 & -2 \end{pmatrix}$

State, without needing to apply Sarrus' rule, the value of its determinant $|B|$. Mention the specific property that has allowed you to reach this conclusion immediately.

In a strategy video game, the attack strength of two units is given by the determinant of the matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$. We know that the value of this determinant is $|A| = -5$.

If the player activates a "booster" that triples only the values of the first row, what will the new value of the determinant be? Justify your answer using the property of multiplying a line by a real number.

Consider two square matrices $X$ and $Y$ of order 3. It is known that $|X| = 4$ and $|Y| = -2$.

Calculate the value of the determinant of the product of both matrices, $|X \cdot Y|$, stating which property you are using.

Without calculating the value of the determinants, justify why the following determinant is equal to zero:

$\begin{vmatrix} 2 & 4 & 8 \\ 1 & 0 & 7 \\ 3 & 6 & 12 \end{vmatrix}$

(Hint: Observe the relationship between the first and third rows).

It is known that the determinant of a matrix $A = \begin{pmatrix} x & y \\ z & t \end{pmatrix}$ is equal to $7$.

From this information, calculate the value of the determinant of the matrix $B = \begin{pmatrix} z & t \\ x & y \end{pmatrix}$. What property have you applied to know whether the sign changes or stays the same?

Given a matrix $C$ of order 3 with a column consisting entirely of zeros:

$C = \begin{pmatrix} 4 & 0 & 1 \\ -1 & 0 & 5 \\ 9 & 0 & 2 \end{pmatrix}$

Reason, without using Sarrus' rule, why the determinant of this matrix is necessarily zero.

In a Mathematics Applied to the Social Sciences exam, Lucas must calculate the determinant of the matrix $3M$ and that of the inverse matrix $M^{-1}$, knowing that $M$ is a square matrix of order $2$ with $|M| = 6$. Lucas writes the following answer:

“Since $|M| = 6$, multiplying by $3$ gives $|3M| = 3 \cdot |M| = 3 \cdot 6 = 18$. On the other hand, since the exponent $-1$ indicates the opposite, the determinant of the inverse changes sign: $|M^{-1}| = -|M| = -6$”.

1. a) Locate and explain with reasoned arguments the two conceptual errors made by Lucas, stating which property of determinants he contradicts in each case.
2. b) Carry out the correct calculations to obtain the true value of $|3M|$ and of $|M^{-1}|$.

A financial consulting firm works with a matrix $A$ of order $3$ whose determinant is $|A| = -4$. To simulate different economic scenarios, two consecutive modifications are performed on $A$: first, the first and second rows are swapped, and then the third row is multiplied by $-2$, resulting in a new matrix $B$.

1. a) Before calculating the numerical value, reasonably estimate the sign of $|B|$: will it be positive or negative? Justify your prediction by analysing the qualitative effect that each of the two transformations has on the original determinant.
2. b) Calculate with precision the numerical value of $|B|$ by applying the corresponding properties step by step.
3. c) Check whether the result obtained confirms your initial estimation and explain why the absolute value of the final determinant has doubled compared to the initial one.

Let $A$ be a square matrix of order 3 whose determinant is $|A| = -2$. Using only the properties of determinants, calculate the value of the following determinants, justifying each step:

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

1. b) $|3A|$.

1. c) $|A^{-1}|$, assuming that $A$ is non-singular.

1. d) The determinant of the matrix resulting from swapping the first and third rows of $A$.

Given two square matrices $A$ and $B$ of order 3, it is known that $|A| = 5$ and $|B| = -1$. Calculate, if possible, the value of the determinant of the following matrices:

1. a) $A \cdot B^2$

1. b) $(A \cdot B)^t$

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

1. d) $A + B$

Without expanding the determinant using Sarrus's rule, justify with reasoning why the value of the following determinant is equal to zero:

$ \begin{vmatrix} a & b & c \\ 2a+3 & 2b+3 & 2c+3 \\ 1 & 1 & 1 \end{vmatrix} $

A logistics company organises its routes between three main cities ($C_1, C_2, C_3$). The traffic volume between them is represented by a matrix $M$ of order 3. Given that the determinant of this matrix is $|M| = 10$, calculate the value of the determinant of a new matrix $N$ obtained by performing the following operations on $M$:

1. The second row was multiplied by $5$.

2. Three times the first row was added to the third row.

3. The first and second columns were swapped.

Given that the value of the determinant $ \begin{vmatrix} x & y & z \\ 1 & 0 & 3 \\ 2 & -1 & 1 \end{vmatrix} = 4$, use the properties of determinants to calculate the value of:

$ \begin{vmatrix} 3x & 3y & 3z \\ 2 & 0 & 6 \\ 2+x & -1+y & 1+z \end{vmatrix} $

Three young entrepreneurs analyze the impact of their social media. The influence matrix $I$ has a determinant $|I| = k$. They discover that if they double the advertising budget (which is equivalent to multiplying the whole matrix by $2$), the new determinant is $|2I| = 160$.

1. a) If the influence matrix is of order $3 \times 3$, calculate the original value of the determinant $k$.

1. b) If the matrix were of order $2 \times 2$ and the determinant of $2I$ were still $160$, would the value of $k$ change? Calculate it.

Justify your answers by applying the property of the product of a matrix by a scalar: if $A$ is of order $n$, then $|t \cdot A| = t^n \cdot |A|$.

Find the values of the parameter $a$ for which the following matrix $A$ has no inverse, using the properties to simplify the calculation of the determinant before setting it equal to zero:

$A = \begin{pmatrix} 1 & a & a^2 \\ 1 & 2 & 4 \\ 1 & 3 & 9 \end{pmatrix}$

(Hint: Try to get zeros in the first column before expanding).

Determine whether the following statements are true or false, reasoning your answer using the properties of determinants:

1. a) If a matrix $A$ has a row that is double another row, then its determinant is always double the determinant of the identity matrix.

1. b) For any square matrix $A$, it holds that $|-A| = -|A|$.

1. c) If $|A| = 0$, then at least one of the rows of the matrix $A$ is a linear combination of the others.

1. d) The determinant of the product of a matrix and its inverse is always 1.

A team from the Institute of Statistics models the flow of goods between three autonomous communities (Galicia, Castilla y León, and Asturias) using a square matrix $T$ of order $3$, whose rows represent the origin of the goods and whose columns represent the destination (in millions of euros). Preliminary analysis shows that the determinant of this matrix is $|T| = 15$.

1. a) Due to tax harmonisation, a uniform tax surcharge of $10\,\%$ is applied to all interregional transactions, so each element of the matrix is multiplied by $1,1$, yielding the matrix $T_1 = 1,1 \cdot T$. Use the properties of determinants to calculate $|T_1|$ and interpret why the determinant does not increase by only $10\,\%$.
2. b) A regional emergency plan exclusively subsidises production in Asturias, managing to double all the values of the third row (exports from Asturias), while the other rows remain untouched, generating the matrix $T_2$. Determine the value of $|T_2|$, stating the property used.
3. c) In a crisis scenario, it is observed that the export balance of Galicia (row 1) becomes a linear combination of the other two ($F_1 = 0,6 \cdot F_2 + 0,4 \cdot F_3$). Justify, without needing to know the specific numbers, what the determinant of the matrix will be in this scenario and what implication it has for the invertibility of the system.

It is desired to construct a matrix expression $B$ from a given matrix $A$ of order $3$ whose determinant is $|A| = -2$.

1. a) Design an expression for $B$ of the form $B = k \cdot (A^t)^m \cdot A^{-1}$, determining with reasoned arguments an integer $k > 0$ and an exponent $m \in \{1, 2\}$ such that the resulting determinant is exactly $|B| = -54$. Show the algebraic development by applying the properties of the determinant of a product, of the transpose, of the inverse, and of the product by a scalar that validate your choice.
2. b) Construct an explicit example of a $3 \times 3$ matrix $C$, containing no zero elements ($c_{ij} \neq 0$) and where no row is a direct multiple of another, but whose determinant is $|C| = 0$. Justify why its determinant is zero by identifying precisely the linear combination existing among its rows.

Let $A$ be a square matrix of order $3$ such that its determinant is $|A| = \frac{1}{2}$. Using only the properties of determinants, calculate with reasoning the value of the determinant of matrix $B$, knowing that:

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

Where $A^t$ is the transpose matrix of $A$ and $A^{-1}$ is its inverse matrix. Justify each step by citing the property applied.

Knowing that the value of the determinant of matrix $M$ is $|M| = k$, calculate, without expanding using Sarrus's rule, the value of the determinant of matrix $N$ in terms of $k$:

$M = \begin{vmatrix} a & b & c \\ p & q & r \\ x & y & z \end{vmatrix} = k \quad \quad N = \begin{vmatrix} p+x & q+y & r+z \\ 3a & 3b & 3c \\ -2x & -2y & -2z \end{vmatrix}$

A logistics company organise their routes between three main warehouses ($A_1, A_2, A_3$). The flow of goods can be represented by a matrix $C$ where the element $c_{ij}$ indicates the transport capacity between warehouse $i$ and warehouse $j$. It is known that the matrix is:

$C = \begin{pmatrix} 1 & m & 0 \\ m & 1 & m \\ 0 & m & 1 \end{pmatrix}$

Determine, using the concepts of determinant and rank, for which values of the parameter $m$ the company's transport structure exhibits redundancies (that is, the routes are linearly dependent and the rank is less than 3).

Find all real solutions to the following equation, using the properties of determinants to simplify the expression before evaluating it:

$\begin{vmatrix} 1 & 1 & 1 \\ x & a & b \\ x^2 & a^2 & b^2 \end{vmatrix} = 0$

Suppose that $a$ and $b$ are distinct real numbers ($a \neq b$). Explain the relationship between the solutions found and the linear dependence of the rows of the matrix.

Consider the matrix $A = \begin{pmatrix} 1 & 0 & -1 \\ 0 & m & 3 \\ 4 & 1 & -m \end{pmatrix}$.

1. a) Determine for which values of $m$ the matrix $A$ has no inverse.
2. b) For $m = 2$, calculate the determinant of the adjugate matrix of $A$, that is, $|Adj(A)|$, by applying properties without explicitly calculating the adjugate matrix.

Consider a square matrix $X$ of order $3$ satisfying the following matrix equation:

$X^2 - 3X = 0$

Prove with clear reasoning that the determinant of $X$ can only take two possible values. What are they? Justify whether a matrix satisfying this condition is necessarily singular.

Given the matrix $B = \begin{pmatrix} k & 1 & 1 \\ 1 & k & 1 \\ 1 & 1 & k \end{pmatrix}$:

1. a) Calculate its determinant by simplifying using elementary row or column operations.
2. b) Determine the rank of $B$ according to the values of the real parameter $k$. Analyse in detail the case in which the rank of the matrix is equal to $1$.

Solve the following system of matrix equations for $X$ and $Y$, using determinants to verify the existence of the required inverses:

$\begin{cases} AX + Y = B \\ X + AY = C \end{cases}$

Where $A = \begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}$, $B = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}$ and $C = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}$.

Note: You must rearrange for $X$ and $Y$ in terms of $A, B$ and $C$ before carrying out the numerical calculations.

To calculate the determinant of the family of matrices $M(x) = \begin{pmatrix} x+2 & x & x \\ x & x+2 & x \\ x & x & x+2 \end{pmatrix}$ and study for which values of $x$ the matrix has no inverse, two students propose different methods:

• Method 1 (Direct by Sarrus' rule): Fully expand the six products of Sarrus' rule, combine like terms to obtain a third-degree polynomial, and factor it using Ruffini's rule.

• Method 2 (Properties of determinants): Add the other two rows to the first row ($F_1 \leftarrow F_1 + F_2 + F_3$), factor out the resulting common factor from the determinant, and then subtract the first column from the other two to make the matrix triangular.

1. a) Apply both procedures step by step until obtaining the fully factored expression of $|M(x)|$.
2. b) Critically compare the computational load and the propensity for algebraic error in each method. Justify with reasoned arguments why Method 2 is superior and how each method would behave if the matrix were of order $4 \times 4$ or higher.
3. c) From the factorisation obtained, determine with reasoned arguments the real values of $x$ for which the matrix $M(x)$ is singular (not invertible).

Let $A$ and $B$ be two square matrices of order $3$ such that neither of them is the zero matrix. It is known that both satisfy the matrix identity:

$$A \cdot B = -2 A^t$$

and it is known that the matrix $A$ is regular ($|A| \neq 0$).

1. a) Formally prove, using exclusively the properties of determinants, that the matrix $B$ is necessarily invertible and calculate the exact numerical value of $|B|$.
2. b) A student claims in their submission: “Since $A \cdot B = -2 A^t$, we can conclude that $|A + B| = |A| + |B|$ by applying determinants to each term”. Rigorously refute this claim by explaining the theoretical fallacy committed, and provide a simple numerical counterexample with diagonal matrices of order $2$ where it is evident that $|X + Y| \neq |X| + |Y|$.
3. c) Knowing that $|A| = 3$, calculate the numerical value of the determinant of the matrix $C = (2 B^{-1} \cdot A)^t$, explicitly justifying each of the properties used in the simplification.