Exercises: Matrix algebra using determinants

In an archaeological excavation, the coordinates of three key points forming a search sector have been recorded. To calculate the area of this sector, we first need to obtain the value of the determinant of the following position matrix $A$:

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

Calculate the value of $|A|$ using Sarrus's rule.

A courier company uses a matrix-based encryption system. For a message to be "decryptable", the matrix used $M$ must be invertible.

Given the matrix $M = \begin{pmatrix} k & 4 \\ 1 & k \end{pmatrix}$, determine for which values of the real parameter $k$ the matrix has no inverse (that is, it is a singular matrix).

Without expanding the matrix completely, apply the properties of determinants to justify why the following determinant is equal to $0$:

$|A| = \begin{vmatrix} 1 & 2 & 3 \\ 5 & 10 & 15 \\ 0 & 1 & -4 \end{vmatrix}$

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

Let $A$ be a square matrix of order $3$ whose determinant is $|A| = -2$. Use the properties of determinants to calculate the following values:

1. a) $|A^t|$ (determinant of the transpose).
2. b) $|A^{-1}|$ (determinant of the inverse).
3. c) $|3A|$ (determinant of the matrix multiplied by the scalar $3$).

Calculate the inverse matrix $A^{-1}$ of the matrix $A = \begin{pmatrix} 3 & 2 \\ 5 & 4 \end{pmatrix}$ using the adjugate matrix method:

$A^{-1} = \frac{1}{|A|} \cdot (\text{Adj}(A))^t$

Follow these steps:

1. Find the determinant $|A|$.

2. Calculate the matrix of cofactors $\text{Adj}(A)$.

3. Take the transpose and divide by the determinant.

Determine the rank of the following matrix $B$ by studying its minors:

$B = \begin{pmatrix} 1 & 2 & 3 \\ 0 & 1 & 1 \\ 2 & 4 & 6 \end{pmatrix}$

Give reasons to state whether the rank is $1$, $2$ or $3$.

Solve the following matrix equation to find the unknown matrix $X$:

$A \cdot X = B$

Where:

$A = \begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 3 \\ 1 & 4 \end{pmatrix}$

(Remember: $X = A^{-1} \cdot B$. You must calculate $A^{-1}$ first).

Calculate the value of the determinant of the following matrix $4 \times 4$ expanding by the elements of the second row (the one with the most zeros):

$|D| = \begin{vmatrix} 1 & 3 & 0 & 5 \\ 0 & 2 & 0 & 0 \\ -1 & 1 & 4 & 2 \\ 2 & 0 & 1 & 3 \end{vmatrix}$

In a Mathematics II exam, Mateo solves the following exercise:

"Knowing that $A$ is a square matrix of order $3$ whose determinant is $|A| = -2$, calculate with justification the value of $|-2\cdot A^{-1}|$ and of $|\operatorname{Adj}(A)|$."

The working Mateo writes is the following:

1) For the first determinant: $|-2\cdot A^{-1}| = -2\cdot |A^{-1}| = -2\cdot \frac{1}{|A|} = -2\cdot \frac{1}{-2} = 1$.

2) For the second determinant: Since the property $A\cdot (\operatorname{Adj}(A))^t = |A|\cdot I$ holds, taking determinants on both sides I obtain $|\operatorname{Adj}(A)| = |A|\cdot |I| = -2\cdot 1 = -2$.

Carry out the following tasks:

1. a) Locate and explain with justification the two conceptual errors made by Mateo, stating precisely which determinant property he applied incorrectly in each section.
2. b) Correct Mateo's working and calculate the correct values of $|-2\cdot A^{-1}|$ and $|\operatorname{Adj}(A)|$, justifying each step using the corresponding properties.
3. c) Deduce the general formula relating $|\operatorname{Adj}(A)|$ to $|A|$ for any invertible square matrix of order $n$, starting from the fundamental identity $A\cdot (\operatorname{Adj}(A))^t = |A|\cdot I$.

Consider the Vandermonde matrix $V$ associated with the ordered real values $x = -1$, $y = 2$, and $z = 4$:

$$V = \begin{pmatrix} 1 & -1 & 1 \\ 1 & 2 & 4 \\ 1 & 4 & 16 \end{pmatrix}$$

1. a) Before carrying out any algebraic calculation, estimate with justification the sign of $|V|$ relying on the relative position of the three values $x < y < z$ and on the factorised structure of the Vandermonde determinant $(y - x)(z - x)(z - y)$. Justify whether the determinant could ever be zero or negative without altering the order of the rows.
2. b) Calculate the exact numerical value of $|V|$ using the elementary row operations needed to create zeros in the first column, checking whether it matches the result of the factorised expression.
3. c) If from $V$ a new matrix $W$ is constructed by swapping its first and third rows, and subsequently multiplying the second row by $-3$: estimate the sign of the new determinant $|W|$ before calculating it and check your estimate by applying the properties of determinants to the result obtained in section b).

Given that the determinant of a square matrix $A$ of order 3 is $|A| = -2$, use the properties of determinants to calculate, with reasoning, the value of the following determinants:

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

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

1. c) The determinant of the matrix resulting from multiplying the first row of $A$ by $5$, swapping the second and third rows, and transposing the matrix.

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

1. a) Determine the values of the real parameter $k$ for which the matrix $M$ is singular (has no inverse).

1. b) For the value $k = 0$, calculate the inverse matrix $M^{-1}$ using the adjugate matrix method.

Solve the following matrix equation $A \cdot X \cdot B = C$, where the matrices $A$, $B$ and $C$ are:

$A = \begin{pmatrix} 1 & 0 \\ 2 & 1 \end{pmatrix}$, $B = \begin{pmatrix} 3 & 1 \\ 1 & 1 \end{pmatrix}$ and $C = \begin{pmatrix} 2 & 0 \\ -1 & 4 \end{pmatrix}$

Justify beforehand that the matrices $A$ and $B$ are invertible by calculating their determinants.

Consider the matrix $A = \begin{pmatrix} 1 & m & 1 \\ m & 1 & 1 \\ 0 & 2 & 2 \end{pmatrix}$.

1. a) Calculate the determinant of $A$ as a function of the parameter $m$.

1. b) Determine the rank of the matrix $A$ according to the values of $m$.

Given the matrix $A = \begin{pmatrix} 2 & 1 & 0 \\ 0 & 1 & -1 \\ 3 & 0 & 2 \end{pmatrix}$, calculate:

1. a) The determinant of $A$ using Sarrus's rule.

1. b) The matrix of cofactors $\text{Adj}(A)$.

1. c) Check that the property $A \cdot \text{Adj}(A)^t = |A| \cdot I$ holds, where $I$ is the identity matrix of order 3.

It is known that a square matrix $B$ satisfies the equation $B^2 - 3B + 2I = O$, where $I$ is the identity matrix and $O$ is the zero matrix.

1. a) Show that $B$ is a non-singular matrix (has an inverse) and express $B^{-1}$ in terms of $B$ and $I$.

1. b) If $B$ is of order 2 and its determinant is $|B| = 2$, what is the determinant of the adjugate matrix of $B$, $|\text{Adj}(B)|$?

Solve the system of matrix equations to find the matrices $X$ and $Y$:

$\begin{cases} 2X + Y = \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix} \\ X - Y = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix} \end{cases}$

Once found, calculate the determinant of the product matrix $X \cdot Y$.

An electronic components company manufactures three types of circuits: $C_1$, $C_2$ and $C_3$. The production cost of each circuit is determined by the determinant of the material matrix $M$:

$M = \begin{pmatrix} 1 & 1 & 1 \\ x & y & z \\ x^2 & y^2 & z^2 \end{pmatrix}$

1. a) Knowing that $x$, $y$ and $z$ represent the quantities of silicon, copper and tin respectively, show that the determinant of this matrix (Vandermonde) is $(y-x)(z-x)(z-y)$.

1. b) If for a specific circuit the quantities are $x=1$, $y=2$ and $z=4$, is the matrix $M$ invertible? Justify your answer using the result from the previous section.

Consider an invertible square matrix $A$ of order $3$ whose determinant is $|A| = -2$. We wish to determine the value of the determinant of the resulting matrix $M = \operatorname{Adj}\big(2\cdot A^t \cdot A^{-1}\big)$.

To solve this question, two students propose opposing strategies:

• Path A (Constructive): Propose a specific diagonal matrix whose determinant is $-2$ (for example, with elements $1$, $1$, and $-2$ on the diagonal), explicitly calculate its transpose, its inverse, multiply them by $2$, find the cofactors of the resulting matrix to construct its adjugate, and finally calculate its determinant using Sarrus' rule.

• Path B (Structural): Chain together the algebraic properties of determinants (determinant of the product, of the transpose, of the inverse, extraction of the scalar $2$ in dimension $3$, and the property of the determinant of the adjugate matrix).

1. a) Critically analyse Path A: argue why calculating a particular matrix does not constitute a valid proof for any matrix $A$ and what algebraic risks it entails.
2. b) Apply Path B, deducing the simplified expression of $|M|$ in terms of $|A|$ and the order $n=3$, showing in detail which property is involved in each equality.
3. c) Obtain the final numerical value of $|M|$ and explain why this result does not depend on the specific value $|A| = -2$, provided that $|A| \neq 0$.

Red Eléctrica de España (REE) monitors photovoltaic solar energy production in a plant in Castilla-La Mancha at three times during a summer day: at $t_1 = 10\text{ h}$, $16\text{ MW}$ are recorded; at $t_2 = 13\text{ h}$, $25\text{ MW}$ are reached (radiation peak); and at $t_3 = 16\text{ h}$, production drops to $16\text{ MW}$. To fit a continuous quadratic prediction curve of the form $P(t) = a\cdot t^2 + b\cdot t + c$:

1. a) State the model as a matrix system $M\cdot X = Y$, where the vector of unknowns is $X = \begin{pmatrix} c \\ b \\ a \end{pmatrix}$, the vector of constant terms is $Y = \begin{pmatrix} 16 \\ 25 \\ 16 \end{pmatrix}$, and the coefficient matrix $M$ has rows of the form $(1, t_i, t_i^2)$.
2. b) Prove that $M$ is a Vandermonde matrix and calculate its determinant $|M|$ using elementary row operations (subtracting the preceding row from each row), without expanding directly using Sarrus' rule.
3. c) Interpret the mathematical meaning of $|M| \neq 0$: what does this fact guarantee about the existence and uniqueness of the prediction parabola? What would happen to the invertibility of $M$ if a sensor failure duplicated the data collection at the same time ($t_2 = t_1$)?

Consider the matrix of order 4 depending on the real parameter $k$:

$$M = \begin{pmatrix} 1 & 1 & 1 & 1 \\ 1 & 1-k & 1 & 1 \\ 1 & 1 & 1+k & 1 \\ k & k & k & k^2 \end{pmatrix}$$

Using only the properties of determinants, perform the following tasks:

1. a) Find the value of the determinant of $M$ in terms of $k$, giving the result in factorised form.

1. b) Determine the values of $k$ for which the matrix $M$ has no inverse.

1. c) For the case $k = 2$, calculate the value of the determinant of the adjugate matrix of $M$, that is, $|Adj(M)|$, without explicitly calculating the elements of this adjugate matrix.

A logistics company uses an encryption algorithm based on matrix algebra to protect its routes. The security key is a matrix $X$ that satisfies the following equation:

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

where $I$ is the identity matrix of order 2 and the matrices $A$ and $B$ are:

$$A = \begin{pmatrix} 2 & 3 \\ 1 & 2 \end{pmatrix}, \quad B = \begin{pmatrix} 10 & 12 \\ 4 & 14 \end{pmatrix}$$

1. a) Justify, using determinants, that the matrix $A$ is invertible.

1. b) Rearrange the above equation for the matrix $X$.

1. c) Explicitly calculate the matrix $X$.

Given the square matrix of order $n$:

$$D_n = \begin{pmatrix} x & a & a & \dots & a \\ a & x & a & \dots & a \\ a & a & x & \dots & a \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a & a & a & \dots & x \end{pmatrix}$$

1. a) For the case $n = 3$, show that the determinant of the matrix is $|D_3| = (x+2a)(x-a)^2$.

1. b) Generalise the previous result for a matrix of order $n$ and deduce the expression for $|D_n|$ in terms of $x, a$ and $n$.

1. c) If $x = 5$ and $a = 2$, for which values of $n$ would the matrix $D_n$ be singular?

Determine the rank of the matrix $A$ according to the values of the real parameters $a$ and $b$:

$$A = \begin{pmatrix} 1 & a & 1 & 2 \\ 2 & b & 2 & 4 \\ 1 & 1 & a & 2 \end{pmatrix}$$

To carry out the study, use the method of bordered minors and explain in detail under what conditions the rank is 1, 2 or 3.

Let $A$ and $B$ be two square matrices of order 3 such that $|A| = 2$ and $|B| = -1$. Calculate, justifying each step using the properties of determinants, the value of the following determinants:

1. a) $|(A^t \cdot B)^{-1}|$

1. b) $|3 \cdot A^2 \cdot Adj(B)|$

1. c) Given that a matrix $C$ satisfies $C^2 = C$ (idempotent matrix), show that its determinant can only be 0 or 1.

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

1. a) Determine the values of the parameter $m$ for which the matrix $A$ has an inverse.

1. b) For $m = 2$, solve the matrix equation $X \cdot A - 2 \cdot Adj(A) = |A| \cdot I$, where $Adj(A)$ is the adjugate matrix of $A$ and $I$ is the identity matrix.

Hint: Use the fundamental relationship between a matrix and its adjugate: $A \cdot (Adj(A))^t = |A| \cdot I$.

In an air traffic control system, the position of three drones at a given moment is defined by the columns of a matrix $P$. In the simplified model used, the configuration is considered safe only if the determinant of $P$ is non-zero. Given the matrix:

$$P(k) = \begin{pmatrix} k & 1 & 1 \\ 1 & k & 1 \\ 1 & 1 & k \end{pmatrix}$$

1. a) Calculate the values of $k$ for which the drones would be in a "critical" configuration (zero determinant).

1. b) For $k = 0$, check whether the matrix is invertible and, if so, find its inverse $P^{-1}$ using the adjugate matrix method.

1. c) Show that for any value of $k \neq 1$ and $k \neq -2$, $P(k)$ is non-singular.

Solve the following system of matrix equations for the unknown matrices $X$ and $Y$:

$$\begin{cases} 2X + Y = A \\ X - Y = B \end{cases}$$

where the coefficient matrices are:

$$A = \begin{pmatrix} 5 & 1 \\ 11 & 4 \end{pmatrix}, \quad B = \begin{pmatrix} 1 & -1 \\ 1 & 2 \end{pmatrix}$$

Once $X$ and $Y$ have been found:

1. a) Calculate $|X \cdot Y|$ and $|X + Y|$.
2. b) Determine whether the product matrix $X \cdot Y$ is invertible using its determinant.

Consider the family of square matrices of order $3$ depending on two real parameters $a$ and $b$:

$$M(a, b) = \begin{pmatrix} 1 & a & a^2 \\ 1 & b & b^2 \\ 1 & 3 & 9 \end{pmatrix}$$

1. a) Using exclusively properties of determinants, obtain the fully factorised expression of $|M(a, b)|$. Determine the equations of the lines in the Cartesian plane $(a, b)$ that form the set of points where the matrix $M(a, b)$ is not invertible.
2. b) INVERSE PROBLEM: Find all pairs of real numbers $(a, b)$ that simultaneously satisfy the following three conditions:

1. The strict ordering $a < b < 3$ holds.

2. The distance between both parameters is one unit: $b - a = 1$.

3. The matrix satisfies the determinantal equation $|2\cdot (M(a, b))^{-1}| = \frac{4}{3}$.

1. c) For the specific matrix $M$ resulting from the pair of values obtained in the previous section, calculate with justification the determinant of its adjugate matrix, $|\operatorname{Adj}(M)|$, and the determinant of the inverse matrix of its transpose, $|(M^t)^{-1}|$.

Given the matrix depending on two real parameters $a$ and $b$:

$$M(a, b) = \begin{pmatrix} 1 & 1 & a \\ 1 & b & 1 \\ a & 1 & 1 \end{pmatrix}$$

To classify the rank and invertibility of $M(a, b)$ in the real plane $(a, b)$, two students compare their methods of work:

• Alba's method: Apply Gaussian elimination using elementary row operations ($F_2 \to F_2 - F_1$, $F_3 \to F_3 - a\cdot F_1$) to put the matrix in row echelon form and count the number of non-zero pivots.

• Bruno's method: Calculate and factorise the major determinant $|M(a, b)|$ using properties to delineate the rank-3 regions, and subsequently study the loci where $|M(a, b)| = 0$ by analysing minors of lower order.

1. a) Critically compare both methods: explain what drawbacks and risks of bifurcation with parameters in the denominators Alba's method presents compared to the robustness of Bruno's method.
2. b) Applying Bruno's method, prove that the determinant factorises as $|M(a, b)| = (1 - a)\cdot [b\cdot(a + 1) - 2]$. Identify geometrically in the plane $(a, b)$ the line and the hyperbola that form the boundary where the matrix lacks an inverse.
3. c) Prove that at almost all points on this boundary the rank of the matrix is $2$, but there exists a unique singular point $(a_0, b_0)$ in the entire plane where the rank drops to $1$. Determine this point, justifying your answer with the value of all its minors of order 2.

Unit challenge. Is it feasible to model and predict the distribution of the national energy mix?

1. Data collection: Access the Red Eléctrica de España website (REE, e·sios platform) or INE and extract the daily production (in GWh) of three renewable sources for 3 consecutive days to construct a square matrix $A_{3\times 3}$.

2. Analytical study: Calculate $|A|$ using Sarrus' rule and determine the rank of $A$ using minors. Incorporate a climate forecast parameter $t$ into one of the rows ($A(t)$); apply the properties of determinants and calculate the limit $\lim_{t \to 1} |A(t)|$ to identify whether the system gets blocked due to linear dependence.

3. Matrix algebra: Build a daily demand vector $B$ (in GWh) as the expected demand: assign probabilities to different consumption scenarios (for example, probability $p = 0,35$ of a winter peak) and weight the demand of each scenario by its probability. Justify the existence of $A^{-1}$ using determinants, calculate it with the adjugate matrix, and solve the equation $A \cdot X = B$ in a spreadsheet to find the weighting factors $X$ (one per source, i.e. per column of $A$, so that the weighted production covers the demand $B$).

4. Final proposal: Write a one-page report discussing the robustness of the model against real grid fluctuations.

The following will be assessed:

• Correctness and skill in calculating $|A|$, $\text{rg}(A)$, and $A^{-1}$ using determinants and Sarrus' rule.

• Rigorous application of matrix algebra, limits, and properties of determinants.

• Genuine extraction of official data and mathematical justification of the conclusions.