Question

Solve each system of equations using Cramer's Rule if is applicable. If Cramer's Rule is not applicable, write, "Not applicable. \(\left\\{\begin{array}{r}x+2 y-z=0 \\ 2 x-4 y+z=0 \\ -2 x+2 y-3 z=0\end{array}\right.\)

Short answer

The solution is (x, y, z) = (0, 0, 0).

Step-by-step solution

Write the system in matrix form

The given system of equations can be written in matrix form as follows:\[\begin{bmatrix}1 & 2 & -1 \2 & -4 & 1 \-2 & 2 & -3\end{bmatrix}\begin{bmatrix}x \y \z\end{bmatrix} = \begin{bmatrix}0 \0 \0\end{bmatrix}\]

Compute the determinant of the coefficient matrix

Find the determinant of the coefficient matrix \[\begin{vmatrix}1 & 2 & -1 \2 & -4 & 1 \-2 & 2 & -3\end{vmatrix}\]Using cofactor expansion along the first row, the determinant can be calculated as:\[D = 1 \cdot \begin{vmatrix}-4 & 1 \2 & -3\end{vmatrix}- 2 \cdot \begin{vmatrix}2 & 1 \-2 & -3\end{vmatrix}- 1 \cdot \begin{vmatrix}2 & -4 \-2 & 2\end{vmatrix}\]

Calculate the 2x2 determinants

Compute the 2x2 determinants:\[\begin{vmatrix}-4 & 1 \2 & -3\end{vmatrix} = (-4)(-3) - (1)(2) = 12 - 2 = 10\]\[\begin{vmatrix}2 & 1 \-2 & -3\end{vmatrix} = (2)(-3) - (1)(-2) = -6 + 2 = -4\]\[\begin{vmatrix}2 & -4 \-2 & 2\end{vmatrix} = (2)(2) - (-4)(-2) = 4 - 8 = -4\]

Combine the results

Substitute the 2x2 determinants back into the formula:\[D = 1 \cdot 10 - 2 \cdot (-4) - 1 \cdot (-4) = 10 + 8 + 4 = 22\]

Determine if D = 0

Since the determinant of the coefficient matrix is not zero (D = 22), Cramer's Rule is applicable.

Form the matrices D_x, D_y, D_z

Replace each column of the coefficient matrix with the constant vector one at a time to form D_x, D_y, D_z.\[D_x = \begin{vmatrix}0 & 2 & -1 \0 & -4 & 1 \0 & 2 & -3\end{vmatrix}\]\[D_y = \begin{vmatrix}1 & 0 & -1 \2 & 0 & 1 \-2 & 0 & -3\end{vmatrix}\]\[D_z = \begin{vmatrix}1 & 2 & 0 \2 & -4 & 0 \-2 & 2 & 0\end{vmatrix}\]

Calculate the determinants D_x, D_y, D_z

All these determinants evaluate to zero since each matrix has a column of zeros.\[D_x = D_y = D_z = 0\]

Apply Cramer's Rule

Using Cramer's Rule, we find the solutions:\[x = \frac{D_x}{D} = \frac{0}{22} = 0\]\[y = \frac{D_y}{D} = \frac{0}{22} = 0\]\[z = \frac{D_z}{D} = \frac{0}{22} = 0\]

Final Step: State the solution

The solution to the system of equations is (x, y, z) = (0, 0, 0).

Key concepts

systems of linear equations

A system of linear equations consists of multiple linear equations involving the same set of variables. In general, a linear equation is a mathematical statement of equality involving a linear combination of variables and constants. An example of a system of linear equations is:o(list)- x + 2y - z = 0- 2x - 4y + z = 0- -2x + 2y -3z = 0c.The goal when working with systems of linear equations is to find the values of the variables that satisfy all the equations simultaneously.
There are several methods to solve these systems, including substitution, elimination, and matrix methods. One matrix method is Cramer's Rule, which uses determinants to find the solution.
What makes linear systems interesting in mathematics is their application in various fields, from engineering to economics.

matrix determinant

In linear algebra, the determinant is a scalar value that can be computed from the elements of a square matrix. It provides important properties of the matrix and useful information about the linear system represented by the matrix.
The determinant of a 2x2 matrix \(\begin{vmatrix} a & b \ c & d \ \ \end{vmatrix}\) is calculated as
\(ad - bc\).
For larger matrices, such as 3x3, the calculation involves minors and cofactors. For instance, the determinant of \(D = \begin{vmatrix} 1 & 2 & -1 \ 2 & -4 & 1 \ -2 & 2 & -3 \ \ \end{vmatrix}\) can be computed using cofactor expansion along the first row.
The determinant determines whether a matrix has an inverse. If a matrix's determinant is zero, it means the matrix is singular and does not have an inverse. For our system of equations, if the determinant is zero, Cramer's Rule cannot be applied.
In this example, we calculated the determinant as \(D = 22\), which is not zero, making the system solvable with Cramer's Rule.

linear algebra

Linear algebra is a branch of mathematics that studies vectors, vector spaces (or linear spaces), linear mappings, and systems of linear equations. It provides a framework and methods for analyzing linear systems.
Core concepts in linear algebra include:

• Vectors and Matrices
• Determinants
• Inverse of matrices
• Eigenvalues and Eigenvectors

These concepts are fundamental for understanding various phenomena in physics, computer science, engineering, and data science.
Matrix operations such as addition, multiplication, and finding inverses are essential techniques in linear algebra. Understanding these operations helps solve complex problems more efficiently. In the context of our exercise, we use matrices to represent the system of equations and determinants to solve them using Cramer's Rule.Linear algebra’s methods and concepts are crucial for practical applications like computer graphics, machine learning, and optimization problems.