Cramer's rule
So far, to solve a system of three unknowns you have used Gauss to reduce it to triangular form. Cramer's rule gives a direct formula: each unknown is a quotient of two determinants. It does not replace Gauss —there are systems where Gauss is quicker— but it is essential when the system contains a parameter, because it lets you solve for the unknowns without going through every step of the reduction.
Cramer systems: when $|A| \ne 0$
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:…
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:
$\begin{cases} 3x + 2y = 7 \\ 5x - y = 2 \end{cases}$
- a) Calculate the determinant of the coefficient matrix, $|A|$.
- b) Justify whether it is a Cramer-type system.
- 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.
Cramer's rule
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…
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:
$$\begin{cases} x + y + z = 6 \\ x - y + z = 2 \\ x + y - z = 0 \end{cases}$$Cramer's rule with three unknowns
The procedure is identical for a $3\times3$ system: four determinants of order $3$ ($|A|$, $|A_x|$, $|A_y|$, $|A_z|$) instead of three of order $2$. There is more work, but the logic —replace a column and divide— does not change.
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…
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:
- a) $\begin{cases} x + y = 5 \\ 2x + 2y = 10 \end{cases}$
- b) $\begin{cases} x + 2y = 3 \\ 3x - y = 1 \end{cases}$
- c) $\begin{cases} x + y + z = 1 \\ y + z = 2 \\ z = 3 \end{cases}$
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…
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.
Cramer's rule with a parameter
The strength of Cramer's rule shows when there is a parameter: you can solve for one unknown without solving the whole system. You calculate $|A(k)|$ and $|A_i(k)|$ as functions of the parameter, and the quotient gives you the unknown as a simplified algebraic fraction.
Consider the following system of linear equations depending on the real parameter $k$: $$\begin{cases} kx + y + z = 1 \\ x + ky + z = k \\ x + y + kz = k^2 \end{cases}$$…
Consider the following system of linear equations depending on the real parameter $k$:
$$\begin{cases} kx + y + z = 1 \\ x + ky + z = k \\ x + y + kz = k^2 \end{cases}$$- a) Discuss the system using the Rouché-Frobenius theorem according to the values of $k$.
- 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 $k$.
Consider the system of equations: $$\begin{cases} ax + y + z = (a-1)^2 \\ x + ay + z = (a-1) \\ x + y + az = 1 \end{cases}$$ a) Determine the values of $a$ for which…
Consider the system of equations:
$$\begin{cases} ax + y + z = (a-1)^2 \\ x + ay + z = (a-1) \\ x + y + az = 1 \end{cases}$$- a) Determine the values of $a$ for which Cramer's rule cannot be applied to the system.
- b) For $a = 0$, solve the complete system using Cramer's rule exclusively.
- 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.
Cramer's rule in problems and matrix equations
Cramer's rule can also be used to find a single unknown without calculating the others: if you are only asked for $z$, $|A|$ and $|A_z|$ are enough. This is especially useful within larger problems set in a context, where the question only asks about one of the variables.
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)…
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.