Exercises: Determinants of order 3

Calculate the value of the following determinant using Sarrus's rule. Make sure you calculate the products along the main and secondary diagonals paying careful attention to the signs:

$$ \left| \begin{array}{ccc} 1 & 2 & -1 \\ 0 & 3 & 2 \\ 4 & -1 & 1 \end{array} \right| $$

Find the value of the unknown $x$ so that the determinant of the following matrix is zero:

$$ \left| \begin{array}{ccc} x & 0 & 2 \\ 1 & 1 & 3 \\ 0 & 2 & 1 \end{array} \right| = 0 $$

In a triangular matrix (upper or lower), the determinant is simply the product of the elements on its main diagonal. Calculate the determinant of the following matrix representing a company's production costs:

$$ \left| \begin{array}{ccc} 5 & 8 & 12 \\ 0 & 2 & 7 \\ 0 & 0 & 3 \end{array} \right| $$

Calculate the determinant of the following matrix, which represents the number of three types of shares (Technology, Energy, Banking) in three different investment portfolios:

$$ \left| \begin{array}{ccc} 10 & 10 & 10 \\ 2 & 3 & 5 \\ 1 & 0 & 2 \end{array} \right| $$

(Hint: Try taking a common factor out of the first row before applying Sarrus to simplify the numbers).

Carlos is solving a class exercise in which he must calculate the determinant of the matrix $M = \begin{pmatrix} 1 & -2 & 3 \\ 0 & 5 & -1 \\ 2 & 1 & 4 \end{pmatrix}$ by applying Sarrus' rule. In his notebook, he writes down the following expansion:

$$|M| = 1 \cdot 5 \cdot 4 + (-2) \cdot (-1) \cdot 2 + 3 \cdot 0 \cdot 1 - 3 \cdot 5 \cdot 2 + 1 \cdot (-1) \cdot 1 + (-2) \cdot 0 \cdot 4 = 20 + 4 + 0 - 30 - 1 + 0 = -7$$

1. a) Locate the conceptual error that Carlos made when applying Sarrus' formula and explain clearly why his sign setup is incorrect.
2. b) Correct the calculation by performing the appropriate steps and determine whether the matrix $M$ is invertible from the value obtained.

A team of surveyors is delimiting a small triangular recreational plot in the municipality of Teruel. On the cadastral map, the coordinates of the three vertices (measured in decametres) are $A(1, 2)$, $B(7, 2)$, and $C(5, 8)$.

1. a) Before carrying out any matrix calculations, make a direct geometric estimate of the area by reasoning from the length of the horizontal segment $AB$ as the base and the vertical distance from $C$ to that base as the height.
2. b) Set up and calculate the exact area using the formula for the determinant of order 3 with the column of ones.
3. c) Check whether the result matches your initial estimate and reason about what would happen to the value of the determinant if you had placed the coordinates of vertex $B$ in the first row and those of vertex $A$ in the second in the matrix.

A mobile app development company is analysing the impact of three of its star features (A, B and C) in three different markets. The performance data are collected in the following matrix $M$:

$M = \begin{pmatrix} 1 & 2 & 1 \\ 0 & 1 & 3 \\ 2 & 1 & 1 \end{pmatrix}$

Calculate the determinant of the matrix $M$ using the Sarrus rule (write the three products of the positive diagonals and the three of the negative ones) to determine whether the analysis model is consistent (that is, whether its determinant is non-zero).

For a digital security system to be valid, the "signature matrix" $S$ must not be singular (its determinant must be non-zero). Given the matrix:

$S = \begin{pmatrix} 1 & a & 0 \\ 2 & 1 & 1 \\ a & 0 & -1 \end{pmatrix}$

Determine the values of the real parameter $a$ for which the security system would not be valid.

Solve the following quadratic equation set up via a determinant:

$\begin{vmatrix} x & 0 & 1 \\ 1 & x & 2 \\ 0 & 1 & 1 \end{vmatrix} = 5$

A digital map places three logistics points on a coordinate plane. The area of a triangle whose vertices are $(x_1, y_1)$, $(x_2, y_2)$ and $(x_3, y_3)$ can be calculated using the formula:

$Área = \frac{1}{2} \cdot | \det(A) |$, where $A = \begin{pmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{pmatrix}$.

Calculate the area of the triangle formed by the points $P_1(1, 2)$, $P_2(4, 5)$ and $P_3(2, 8)$ shown in the following graph:

To find out for which values of the real parameter $m$ the matrix $M = \begin{pmatrix} 1 & 1 & 1 \\ m & 2 & 3 \\ m^2 & 4 & 9 \end{pmatrix}$ does not have an inverse matrix, two students propose two different strategies:

• Path 1 (mechanical): Apply Sarrus' rule term by term, group the powers of $m$ to obtain a quadratic equation, and solve $|M| = 0$.

• Path 2 (structural): Observe the columns of the matrix and deduce directly which values of $m$ would make two columns identical, applying the property that a determinant with two equal columns vanishes.

1. a) Expand the determinant following Path 1 and calculate the solutions to $|M| = 0$.
2. b) Explain with reasoning how Path 2 allows both solutions to be accurately anticipated without the need to manipulate any polynomial.
3. c) Justify which of the two methods you consider safer in this case and conclude for which values of $m$ the matrix $M$ is non-singular.

In a Galician forest district, environmental rangers supervise a special fire surveillance area delimited by three observation posts. On the cartographic map (with scale in kilometres), the posts are located at positions $N(2, 8)$, $E(7, 5)$, and $S(3, 1)$.

1. a) Model the triangular surface delimited by the three posts using a determinant of order 3 and calculate the area of the protected zone (in $\text{km}^2$).
2. b) There is a plan to install a telecommunications cabin at a point with coordinates $P(x, y)$ that must be located exactly on the straight line connecting the South post $S$ with the North post $N$. Use the condition of a null determinant (zero area or collinearity) to obtain the linear equation that the coordinates $(x, y)$ of this cabin must satisfy.

In the design of a new map for a strategy video game, the vertices of a protected triangular region are given by the coordinates of the points $P(1, 1)$, $Q(3, 5)$ and $R(k, 7)$. Point $R$ can be anywhere on the line $y = 7$, depending on the value of $k$. The game engine requires the area of this region to be exactly $10$ square units in order to balance the factions.

Use the area formula with order-3 determinants to find the possible values of the parameter $k$ that satisfy this condition.

A logistics company analyses the feasibility of three delivery routes. The associated cost matrix $M$ depends on a fuel price fluctuation factor $t$:

$$M = \begin{pmatrix} 1 & t & 0 \\ t & 1 & t \\ 1 & t & t \end{pmatrix}$$

It is known that the logistics system stalls (the matrix is not invertible) when the determinant of $M$ is zero. Determine for which values of $t$ the company will experience route blockages.

Solve the following cubic equation set up using the determinant of a matrix containing logarithmic functions. Find the value of $x$ knowing that $x > 0$:

$$\begin{vmatrix} \log(x) & 0 & 0 \\ 1 & \log(x) & 2 \\ 0 & -1 & 1 \end{vmatrix} = 6$$

(Hint: Expand the determinant and solve the resulting equation for the variable $\log(x)$).

Prove, using the properties of determinants, that the following determinant is zero for any value of the variables $a, b, c$, without needing to apply Sarrus's rule:

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

(Hint: Try performing elementary row operations to obtain proportional elements).

Design a square matrix $A$ of order $3$ that simultaneously satisfies the following four conditions:

1. All its elements belong to the set of integers $\{-1, 0, 1, 2\}$.

2. None of the elements on its main diagonal is zero ($a_{11} \cdot a_{22} \cdot a_{33} \neq 0$).

3. When expanding its determinant using Sarrus' rule, exactly two of the six elementary products are non-zero.

4. The resulting determinant is $|A| = -5$.

1. a) Write the matrix you have constructed, show the explicit expansion of the six Sarrus products identifying which ones vanish, and verify that the determinant is $-5$.
2. b) Reason mathematically why it would be IMPOSSIBLE to construct a matrix satisfying conditions 1, 2, and 3 but whose determinant is $|A| = 11$, analysing the possible numerical values that a product of three factors chosen from the allowed set can take.

A logistics distribution platform in the Zaragoza corridor analyses the operational capacity of three electric van prototypes (A, B, and C). The capacity of each vehicle is described by a three-dimensional vector that includes: useful volume in $\text{m}^3$, maximum payload in hundreds of kilograms, and number of standard containers:

• Prototype A: $\vec{u} = (4, 3, 2)$

• Prototype B: $\vec{v} = (2, 5, 1)$

• Prototype C: $\vec{w} = (k, 1, 3)$, where $k$ is the useful volume in $\text{m}^3$ of this prototype in the design phase.

1. a) Write the matrix whose rows are the three capacity vectors and obtain the expression for its determinant in terms of $k$ by applying Sarrus' rule.
2. b) For the fleet to be efficient, logistics management requires prototype C to provide a truly independent capacity (that is, the three vectors must not be coplanar and the matrix must be non-singular, preventing one of the models from being a redundant linear combination of the other two). Find the value of $k$ that would make prototype C redundant and interpret what this case means with respect to the volume of the parallelepiped of capacities.
3. c) If vehicle C is finally approved with $k = 8$, calculate the resulting determinant, interpret its absolute value geometrically, and justify whether the company will be able to fulfill any requested three-dimensional transport order uniquely and unambiguously.