Exercises: Cramer's rule

To start working with Cramer's rule, it is essential to identify which systems allow its direct use. Given the following system of two equations with two unknowns:

$\begin{cases} 3x + 2y = 7 \\ 5x - y = 2 \end{cases}$

1. a) Calculate the determinant of the coefficient matrix, $|A|$.
2. b) Justify whether it is a Cramer-type system.
3. c) Find the value of the unknown $x$ by calculating only the determinant of the matrix associated with that variable, $|A_x|$, and applying Cramer's formula.

Apply Cramer's rule to solve the following $3 \times 3$ system of linear equations. Make sure you first calculate the general determinant of the system to check that it has a unique solution:

In a Mathematics II exam, students are asked to identify Cramer-type systems without solving them. Give reasons to indicate which of the following systems cannot be solved directly using Cramer's rule:

1. a) $\begin{cases} x + y = 5 \\ 2x + 2y = 10 \end{cases}$

1. b) $\begin{cases} x + 2y = 3 \\ 3x - y = 1 \end{cases}$

1. c) $\begin{cases} x + y + z = 1 \\ y + z = 2 \\ z = 3 \end{cases}$

Imagine you are managing a team's points in a video game. The scoring system depends on three types of missions ($x$, $y$, $z$) and is given by:

$\begin{cases} 2x + y + z = 10 \\ x + 2y + z = 8 \\ x + y + 2z = 12 \end{cases}$

Use Cramer's rule to calculate only the value of $z$, which represents the expert-level missions. It is not necessary to find the value of $x$ or $y$.

Two friends, Lucas and Sofía, went to a shop selling phone cases and screen protectors. Lucas bought 2 cases and 3 screen protectors for 22 €. Sofía bought 3 cases and 2 screen protectors for 23 €.

1. a) Set up the system of equations associated with the problem.
2. b) Use the determinants $|A|$, $|A_x|$ and $|A_y|$ to find the price of a case ($x$) and a screen protector ($y$) using Cramer's rule.

Consider the following homogeneous system:

$\begin{cases} 2x - 3y + z = 0 \\ x + y - 2z = 0 \\ 3x - 2y - z = 0 \end{cases}$

1. a) Calculate the determinant of the coefficient matrix $|A|$.
2. b) Based on the previous result and without calculating any further determinants, explain how many solutions the system has and what main solution always satisfies a system of this type.

In a system of linear equations $AX = B$, we know that the determinant of the coefficient matrix is $|A| = 5$. If we want to find the value of the unknown $y$ and we know that the determinant of the matrix resulting from replacing the second column with the constant terms is $|A_y| = -15$:

1. a) What is the value of the unknown $y$ according to Cramer's rule?
2. b) If we also know that $x = 4$, what must the value of the determinant $|A_x|$ be?

Solve the following system using Cramer's rule. This is a system where one of the unknowns is already "isolated" in the structure, which simplifies the determinant calculations:

$\begin{cases} x + y + z = 4 \\ 3x - y = 1 \\ 3y + 2z = 7 \end{cases}$

Check your result by substituting the obtained values into the first equation.

Marcos wants to solve the following system of two equations with two unknowns by applying Cramer's rule:

To find the value of the unknown $y$, he writes the following steps on his exam:

1. He calculates the determinant of the coefficient matrix: $|A| = \begin{vmatrix} 2 & 5 \\ 3 & -1 \end{vmatrix} = 2 \cdot (-1) - 3 \cdot 5 = -17$. Since $|A| \neq 0$, he correctly states that it is a Cramer's system.

2. To calculate the unknown $y$, he constructs the associated matrix: $|A_y| = \begin{vmatrix} 11 & 5 \\ 8 & -1 \end{vmatrix} = 11 \cdot (-1) - 8 \cdot 5 = -51$.

3. He concludes that $y = \dfrac{|A_y|}{|A|} = \dfrac{-51}{-17} = 3$.

1. a) Identify and explain, with reasoning, the conceptual error that Marcos made when constructing the matrix to calculate $y$.
2. b) Correct his work by calculating the correct determinant $|A_y|$ and find the true value of the unknown $y$.

An agricultural cooperative analyses the yield of three crop plots ($x$, $y$, $z$) whose production in tonnes corresponds to the following system of linear equations:

The board of directors only needs to know the production of plot $z$ to urgently close an export contract.

1. a) Two strategies are proposed: applying the Gaussian elimination method until the system is completely echeloned and solving for all three unknowns, or checking if it is a Cramer's system and calculating exclusively the determinants necessary to obtain $z$. Compare both approaches and justify which one is more direct and computationally economical for this objective.
2. b) Apply the more efficient strategy you have chosen to calculate the exact value of production $z$.

An e-sports team has participated in a tournament where points are awarded for won games ($x$), tied games ($y$), and secondary objectives completed ($z$). The system that describes their score is the following:

$\begin{cases} x + y + z = 10 \\ 2x - y + 3z = 13 \\ x + 2y - z = 4 \end{cases}$

Use Cramer's rule to find how many games they won, how many they tied, and how many secondary objectives they achieved.

Given the system of linear equations depending on the real parameter $k$:

$\begin{cases} kx + y + z = 1 \\ x + ky + z = 1 \\ x + y + kz = 1 \end{cases}$

Determine for which values of $k$ the system is a Cramer system and solve it for $k = 0$.

A municipal shared mobility service uses three types of vehicles: city bikes, e-bikes and scooters. It is known that weekly energy consumption (in hundreds of kWh) satisfies the following relationships:

$\begin{cases} 2x + 3y + z = 11 \\ x - 2y + 3z = 6 \\ 3x + y - 2z = -1 \end{cases}$

Where $x, y, z$ represent the fleet of each vehicle respectively. Calculate the number of units of each type using determinants.

Express the following system in matrix form $A \cdot X = B$ and, after checking that the determinant of the coefficient matrix is nonzero, find the value of the unknown $y$ by applying exclusively the formula of Cramer's rule:

$\begin{cases} x + 2y - z = -1 \\ 3x - y + 2z = 1 \\ 2x + 3y + z = 2 \end{cases}$

A nutritionist designs a diet with three basic foods $A, B$ and $C$. The system of equations relating the portions of each food with total grams of protein, carbohydrates and fats is:

$\begin{cases} x + y + z = 6 \\ 2x + y = 7 \\ 3x + 2y + z = 13 \end{cases}$

Is it possible to apply Cramer's rule directly to solve this system? Justify your answer by analyzing the determinant of the associated matrix and, if so, calculate how many portions of food $z$ should be included.

Three planes in space are given by the equations of the following system:

$\pi_1: x + y + z = 2$

$\pi_2: 2x + 3y + z = 3$

$\pi_3: kx + 10y + 4z = 11$

1. a) Determine for which values of $k$ the three planes intersect at a single point.

1. b) For $k = 1$, find the coordinates of that point using Cramer's rule.

It is known that in a system of three equations with three unknowns $AX=B$, the determinants of the auxiliary matrices are $|A_x| = 12$, $|A_y| = -6$ and $|A_z| = 0$. If it is known that the solution for $x$ is $2$, use the logic of Cramer's rule to determine the value of the determinant of the coefficient matrix $|A|$ and the value of the unknowns $y$ and $z$.

Solve the following system of equations using Cramer's rule, paying special attention to handling negative constant terms:

$\begin{cases} 3x - 2y + z = -10 \\ 2x + y - 3z = 5 \\ x - y + 2z = -5 \end{cases}$

An organic fruit distributor sells three types of weekly baskets: Basic ($x$), Family ($y$), and Gourmet ($z$), whose unit prices in euros satisfy the following accounting relationships:

1. a) Before calculating the solution, look at the last two equations knowing that the prices must be positive quantities. Justify why the price of the Basic basket $x$ must be strictly less than 40 € and reasonably estimate which of the three baskets ($x$, $y$, or $z$) will have a price significantly higher than the other two.
2. b) Prove that the system is a Cramer's system by calculating the determinant of the coefficient matrix, $|A|$.
3. c) Solve the complete system using Cramer's rule and check whether the prices obtained are fully consistent with the bounds and the estimate you made in section (a).

A delegation of students from Zaragoza travels by high-speed train to Madrid to attend a university fair. In total, 24 people travel and they purchase three types of Renfe tickets: Básico ($x$), Elige ($y$), and Prémium ($z$). The following information about the purchase is available:

• The total number of tickets purchased is 24.

• The fares are 25 € for the Básico ticket, 40 € for the Elige, and 60 € for the Prémium, bringing the total invoice amount to 780 €.

• The number of Básico fare tickets purchased exceeds the sum of the tickets of the other two fares combined by 8 units.

1. a) Set up the system of three linear equations with three unknowns that models the purchase of the tickets.
2. b) Justify that it is a Cramer's system by calculating the determinant $|A|$.
3. c) Apply Cramer's rule to determine how many tickets of each type were purchased.

Consider the following system of linear equations depending on the real parameter $k$:

1. a) Discuss the system using the Rouché-Frobenius theorem according to the values of $k$.
2. b) In the case where the system is a Cramer system, solve for the unknown $y$ applying Cramer's rule. Express the result as a simplified algebraic fraction in terms of $k$.

Three digital marketing agencies, Alpha, Beta and Gamma, manage the campaigns for a new video game. It is known that:

• The total cost of the three campaigns was $12\,000$ €.

• The investment in Beta is double the difference between the investment in Gamma and the investment in Alpha.

• If the investment in Alpha were tripled, that in Beta halved, and that in Gamma increased by $1\,000$ €, the total cost would be $19\,000$ €.

1. a) Set up a system of linear equations to calculate how much each agency has invested.
2. b) Show that it is a Cramer system.
3. c) Use Cramer's rule to find the budget allocated to the agency Gamma.

Consider the system of equations:

1. a) Determine the values of $a$ for which Cramer's rule cannot be applied to the system.
2. b) For $a = 0$, solve the complete system using Cramer's rule exclusively.
3. c) Is there any value of $a$ for which the value of the unknown $z$ is equal to $0$ while the system is consistent determined? Justify your answer.

A logistics company uses three types of vans ($A$, $B$ and $C$) to deliver parcels. On Monday they used $4$ of type $A$, $1$ of type $B$ and $3$ of type $C$ to deliver $130$ parcels. On Tuesday, with $2$ of type $A$, $5$ of type $B$ and $1$ of type $C$, they delivered $110$ parcels. On Wednesday, they delivered $120$ parcels using $3$ vans of each type.

1. a) Write the coefficient matrix and calculate its determinant.
2. b) Knowing that the system is consistent determined, what is the load capacity of the type $B$ van? Solve it using Cramer's rule.
3. c) If a fourth van of type $D$ has a capacity equal to the sum of the capacities of $A$ and $C$ minus that of $B$, how many parcels can it transport?

Let the system of equations be:

1. a) Calculate the determinant of the coefficient matrix and reason why it is always a Cramer system, regardless of the values of $m, n$ and $p$.
2. b) Applying Cramer's rule, show that the solution for $x$ is $x = \frac{m+n}{2}$.
3. c) If it is known that the solution of the system is $x = 5, y = 1, z = 2$, determine the values of the constant terms $m, n$ and $p$.

An investor divides $50\,000$ € between three financial products: a savings account at $1\%$, an investment fund at $3\%$ and tech shares at $7\%$. At the end of the year, total profits were $2\,300$ €. It is also known that the amount invested in tech shares is equal to three times the amount invested in the savings account plus $2\,000$ €.

1. a) Set up the system of equations to determine the three amounts invested ($x, y, z$).
2. b) Write the expression of Cramer's rule to find the amount invested in the investment fund ($y$). It is not necessary to perform the final calculations of the determinants of $3 \times 3$, just leave them set up.
3. c) If the determinant of the system matrix is $\Delta = -0,10$, is it possible for the system to be inconsistent? Reason your answer.

Given the system of equations depending on parameter $a$:

1. a) Study for which values of $a$ the system is consistent determined.
2. b) For the values found in the previous section, apply Cramer's rule to find the value of $z$ in terms of $a$.
3. c) Determine for which value of $a$ it holds that $x=y=z$.

In a sports nutrition study, the composition of three supplements ($S_1, S_2, S_3$) in grams of protein, carbohydrates and fats per unit is analyzed:

An athlete needs to consume exactly $115$ g of protein, $140$ g of carbohydrates and $34$ g of fats.

1. a) Set up the system of equations to determine how many units of each supplement he should take.
2. b) Calculate the determinant of the coefficient matrix. What does this value indicate about the possibility of using Cramer's rule?
3. c) Solve the system using Cramer's rule to determine how many units of supplement $S_1$ he must consume.

In a linear system of three equations with three unknowns expressed in matrix form as $A \cdot X = B$, the columns of the coefficient matrix are $C_1$, $C_2$, and $C_3$, such that $A = (C_1 \; C_2 \; C_3)$. It is known that the determinant of the coefficient matrix is $|A| = 4$ and that the column of constant terms satisfies the vector relation $B = 3 C_1 - 2 C_2 + 5 C_3$.

1. a) Without needing to know the individual numerical values of column $C_1$, deduce and justify the unique solution $(x, y, z)$ of the system, relying on the theoretical foundation of Cramer's rule explained in the notes (the decomposition of $B$ as a linear combination of columns).
2. b) Find the exact numerical value of the three determinants $|A_x|$, $|A_y|$, and $|A_z|$ involved in Cramer's rule for this system.
3. c) If the column of constant terms is replaced by a new column $B' = C_1 + k \cdot C_2$, determine with reasoning what value the real parameter $k$ must take so that the determinant associated with the unknown $y$ is exactly $|A_y| = -12$.

Two 2nd year Bachillerato students analyse the following system of linear equations dependent on the real parameter $a$:

• Carlos says: «We should apply Gaussian elimination from the start, creating zeros with the parameter $a$, because that way we discuss and calculate the solution at the same time».

• Diana argues: «Since there is a parameter, Gauss forces you to divide by expressions containing $a$ and open multiple premature cases; it is much safer to calculate $|A(a)|$, apply Cramer's rule to obtain the general solutions as algebraic fractions when $|A(a)| \neq 0$, and leave Gauss only for the critical values where the determinant vanishes».

1. a) Calculate the polynomial $|A(a)|$ and determine for which values of $a$ the system is a Cramer's system.
2. b) For the values where the system is a Cramer's system, use Cramer's rule to calculate the unknown $x$ in terms of $a$. Simplify the resulting algebraic fraction by factoring the numerator and denominator.
3. c) Critically assess the arguments of Carlos and Diana: explain the operational difficulties presented by the Gaussian elimination method compared to Cramer's rule when seeking the general solution of a determinate consistent system with parameters.