Solving systems and Cramer's rule
Until now, to solve a system with three unknowns you used Gauss to reduce it to triangular form. Cramer's rule offers 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 following every step of the triangulation.
Cramer system: when $|A| \ne 0$
A nutritionist designs a personalised diet for an athlete by combining three types of food ($A, B$ and $C$). The amounts of protein, carbohydrates and fats per unit of…
A nutritionist designs a personalised diet for an athlete by combining three types of food ($A, B$ and $C$). The amounts of protein, carbohydrates and fats per unit of food are given by the following system of equations, where $x, y, z$ represent the units of each food required to reach the daily targets:
$\begin{cases} 2x + y + z = 7 \\ x + 2y + z = 8 \\ x + y + 2z = 9 \end{cases}$
- a) Write down the coefficient matrix $A$ and the augmented matrix $A^*$.
- b) Verify that the system is a Cramer system by calculating the determinant of $A$.
- c) Find the value of the unknown $x$ applying exclusively Cramer's rule.
Cramer's rule
Two friends, Lucía and Marcos, are comparing their subscriptions to streaming platforms. Lucía pays a total of $16$ € per month for one "Cinema" plan and two "Series"…
Two friends, Lucía and Marcos, are comparing their subscriptions to streaming platforms. Lucía pays a total of $16$ € per month for one "Cinema" plan and two "Series" plans. Marcos, for his part, has two "Cinema" plans and one "Series" plan for $17$ € per month. The figure summarises the plans each of them has.
- a) Express the situation as a system of two equations with two unknowns (let $x$ be the monthly price of the Cinema plan and $y$ that of the Series plan).
- b) Calculate the determinant of the coefficient matrix.
- c) Solve the system using Cramer's Rule to find the monthly price of each plan.
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 $2$. The workload grows, but the logic —replace a column and divide— does not change.
In an esports tournament, points are awarded for matches won ($x$), drawn ($y$) and for secondary objectives completed ($z$). A team obtained the following results over…
In an esports tournament, points are awarded for matches won ($x$), drawn ($y$) and for secondary objectives completed ($z$). A team obtained the following results over three matchdays:
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Matchday 1: $1$ win, $1$ draw and $1$ secondary objective = $10$ points.
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Matchday 2: $2$ wins and $1$ draw = $13$ points.
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Matchday 3: $1$ win and $2$ secondary objectives = $11$ points.
- a) Set up the associated system of equations.
- b) Solve it using Cramer's rule to determine how many points each category is worth.
Solve the following system of linear equations using Cramer's rule: $\begin{cases} 2x - y + z = 3 \\ x + 2y - z = 4 \\ 3x - y + 2z = 8 \end{cases}$ To do this, first…
Solve the following system of linear equations using Cramer's rule:
$\begin{cases} 2x - y + z = 3 \\ x + 2y - z = 4 \\ 3x - y + 2z = 8 \end{cases}$
To do this, first identify the coefficient matrix $A$ and calculate its determinant. Then, obtain the determinants of the auxiliary matrices $A_x$, $A_y$ and $A_z$ to find the values of the unknowns.
Cramer's rule in problems and matrix equations
Cramer's rule is also useful for finding 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, such as when solving for a column in a matrix equation.
Given the matrices $A = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 1 \\ m & 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 1 \\ 0 & 1 \\ 1 & 0 \end{pmatrix}$: a) Determine…
Given the matrices $A = \begin{pmatrix} 1 & 0 & 2 \\ 0 & 1 & 1 \\ m & 0 & 1 \end{pmatrix}$ and $B = \begin{pmatrix} 1 & 1 \\ 0 & 1 \\ 1 & 0 \end{pmatrix}$:
- a) Determine the values of the real parameter $m$ for which the matrix $A$ is invertible.
- b) For $m = 0$, solve the matrix equation $A \cdot X + B = 2B$, where $X$ is a matrix of appropriate dimensions.
- c) Set up a system of equations to calculate the first column of the matrix $X$ and solve it using Cramer's rule.