Exercises: Determinants of order 2

Calculate the value of the following 2nd-order determinants, paying special attention to the use of negative signs:

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

1. b) $\begin{vmatrix} -1 & -6 \\ 0 & 7 \end{vmatrix}$

1. c) $\begin{vmatrix} 10 & 15 \\ 2 & 3 \end{vmatrix}$

Find the value of the determinant of the matrix $A$ when it contains simple fractions. This type of calculation is common when working with allocation coefficients in economics problems:

$A = \begin{pmatrix} \frac{1}{2} & 4 \\ \frac{1}{4} & 6 \end{pmatrix}$

Determine the value of the unknown $x$ so that the following equality holds, where the result of the determinant is a known integer value:

$\begin{vmatrix} x & 5 \\ 2 & 8 \end{vmatrix} = 14$

A determinant of order 2 can represent the area of a parallelogram defined by two vectors in the plane. Calculate the absolute value of the determinant formed by the vectors $\vec{u} = (4, 1)$ and $\vec{v} = (2, 5)$ to find that area:

$Área = \left| \begin{vmatrix} 4 & 2 \\ 1 & 5 \end{vmatrix} \right|$

Marcos is analyzing the invertibility of the matrix $A = \begin{pmatrix} 2 & -4 \\ -3 & 6 \end{pmatrix}$ and writes the following in his notebook:

From this value, he concludes that the matrix is non-singular and therefore has an inverse matrix.

1. a) Locate and explain in detail the calculation error involving signs that Marcos made in the expansion of the determinant.
2. b) Correct the calculation and explain whether Marcos's final conclusion regarding the existence of the inverse was correct or incorrect.

An agricultural estate in La Rioja has the shape of a parallelogram defined by the vectors $\vec{u} = (5, 1)$ and $\vec{v} = (1, 4)$, whose components are measured in hectometres.

1. a) Reasonably estimate between which values the area must lie: if you mentally construct an outer rectangle that encloses the plot with sides parallel to the axes measuring $6$ and $5\text{ hm}$, can the area of the plot exceed $30\text{ hm}^2$? Justify why it must be strictly less.
2. b) Check your estimate by calculating the exact area using the 2nd-order determinant associated with the two vectors, and analyse whether the numerical result obtained is consistent with your previous reasoning.

Consider the matrix $A = \begin{pmatrix} x+1 & 2 \\ x & x-1 \end{pmatrix}$. Determine the real values of $x$ for which the determinant of the matrix is equal to $6$.

In designing a logo for a new tech startup, a designer uses a parallelogram defined by the vectors $\vec{u} = (a, 1)$ and $\vec{v} = (2, a)$. It is known that the area of this parallelogram is given by the modulus of the determinant of the matrix formed by both vectors.

Calculate the value of the parameter $a$ so that the area of the logo is $3$ square units.

Show, without calculating the final numerical value, that the following determinant is zero for any value of the angle $\alpha$, using basic trigonometric identities:

Solve the following equation involving determinants of order 2:

A courier company based in Zaragoza bills two shipping methods: standard ($x$ €) and express ($y$ €). During the first half of a month, it handles $120$ standard shipments and $80$ express shipments. In the second half, after launching a marketing campaign, the standard shipments multiply by a factor $k$ (resulting in $120k$ shipments) and the express shipments double exactly ($160$ shipments). The activity matrix for the month is:

1. a) Determine for which value of the factor $k$ the matrix $M$ is singular (its determinant is zero).
2. b) In the business context, if the matrix is singular, is it possible to uniquely deduce the individual rates $x$ and $y$ knowing only the total revenue of each fortnight? Explain what proportional relationship the shipments of the second fortnight have with respect to the first in that case.

Design a square matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ of order 2 that simultaneously satisfies the following three conditions:

1. Its four elements ($a, b, c, d$) must be non-zero integers that are distinct from each other.

2. The sum of the elements on its main diagonal must be equal to $3$ ($a + d = 3$).

3. Its determinant must be exactly $|A| = -5$.

Show step by step the algebraic and deductive procedure you followed to find the four values and justify why the matrix you obtained is non-singular.

In a market study on the elasticity of two competitive products, a coefficient matrix $M$ depending on an investment variable $x$ has been obtained:

Determine for which investment values $x$ the determinant of the matrix is zero, which would indicate a critical point of instability in the model. Remember that, due to the nature of the study, $x$ must be a positive real number.

A graphic designer is working with vector editing software. To transform a figure in the plane, the program uses 2nd-order transformation matrices. The deformation matrix is given by:

Prove, using trigonometric identities, that the determinant of this matrix is constant and independent of the value of the angle $x$. What geometric interpretation does the fact that $|A| = 1$ have in terms of the area of the transformed figure?

Given the matrix depending on the real parameter $m$:

1. a) Calculate the value of its determinant in terms of $m$.
2. b) Determine all values of $m$ for which the matrix does not have an inverse (singular matrix).
3. c) If $m = 0$, calculate the determinant of the inverse matrix $A^{-1}$ using the properties of determinants, without finding the inverse matrix explicitly.

In a matrix encryption system, a key matrix $K$ of order 2 is used. For the system to be secure and allow decryption, the matrix $K$ must be invertible. The following matrix is proposed:

1. a) Calculate the values of $k$ that make the system insecure (that is, such that $K$ has no inverse).
2. b) If it is known that the determinant of the key matrix must be equal to $10$ to comply with a specific security standard, calculate the possible values of $k$.

To find out which values of the real parameter $t$ make the matrix

singular (having no inverse), two students propose different strategies:

• Method 1 (Polynomial expansion): Multiply the diagonals $(t^2 - 1)\cdot 4 - (t + 1)(2t - 2)$, expand all the products, simplify the resulting quadratic polynomial, and set it equal to zero.

• Method 2 (Prior factoring): Note that $t^2 - 1 = (t-1)(t+1)$ and $2t - 2 = 2(t-1)$, and factor out a common term in the expression for $|M(t)|$ before performing extensive multiplications.

1. a) Apply both methods step by step until you find the values of $t$ that make the matrix singular.
2. b) Compare the two approaches: explain which one minimizes the risk of making sign errors with the subtrahend and justify which one you would recommend using in a university entrance exam if time is limited.

A consortium is designing a wind farm on a parallelogram-shaped plot of land in Castilla-La Mancha. Taking the substation as the origin $(0,0)$, two consecutive sides are described by the vectors $\vec{u} = (x, 2)$ and $\vec{v} = (3, x-1)$, whose coordinates are expressed in kilometres, with the technical restriction that $x > 0$.

1. a) Write the algebraic expression of the determinant of the matrix formed by the vectors $\vec{u}$ and $\vec{v}$ in terms of $x$.
2. b) If the Environmental Impact Statement requires the area of the wind farm to be exactly $6\text{ km}^2$, set up the corresponding equation using the absolute value and prove that there are exactly two acceptable values for $x$.
3. c) Calculate the value of the determinant for each of the two solutions found and interpret geometrically why in one case the determinant turns out positive and in the other negative, clarifying why both configurations provide perfectly valid areas.