In Exercises 37–44, use Cramer's Rule to solve each system. {x+y+z=02x−y+z=−1−x+3y−z=−8\begin{cases} x + y + z = 0 \\ 2x - y + z = -1 \\ -x + 3y - z = -8 \end{cases}⎩⎨⎧x+y+z=02x−y+z=−1−x+3y−z=−8

Ch. 6 - Matrices and Determinants

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

문제 37

7장, 문제 37

In Exercises 37–44, use Cramer's Rule to solve each system. $\begin{cases}\) x + y + z = 0 \\ 2x - y + z = -1 \\ -x + 3y - z = -8 \(\end{cases}$

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Write the system of equations in matrix form as $A\mathbf{x} = \mathbf{b}$, where $A$ is the coefficient matrix, $\mathbf{x}$ is the column vector of variables, and $\mathbf{b}$ is the constants vector. For this system, $A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & -1 & 1 \\ -1 & 3 & -1 \end{bmatrix}$, $\mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}$, and $\mathbf{b} = \begin{bmatrix} 0 \\ -1 \\ -8 \end{bmatrix}$.

Calculate the determinant of the coefficient matrix $A$, denoted as $\det(A)$. This determinant must be non-zero to apply Cramer's Rule.

Form three new matrices $A_x$, $A_y$, and $A_z$ by replacing the respective columns of $A$ with the constants vector $\mathbf{b}$. Specifically, $A_x$ replaces the first column, $A_y$ replaces the second column, and $A_z$ replaces the third column with $\mathbf{b}$.

Calculate the determinants of these new matrices: $\det(A_x)$, $\det(A_y)$, and $\det(A_z)$.

Use Cramer's Rule to find the solutions for $x$, $y$, and $z$ by dividing each determinant by $\det(A)$: $x = \frac{\det(A_x)}{\det(A)}$, $y = \frac{\det(A_y)}{\det(A)}$, and $z = \frac{\det(A_z)}{\det(A)}$.

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주요 개념

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Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. It applies when the system has the same number of equations as unknowns and the determinant of the coefficient matrix is non-zero. Each variable is found by replacing the corresponding column in the coefficient matrix with the constants vector and calculating the determinant ratio.

Determinants of Matrices

The determinant is a scalar value that can be computed from a square matrix and provides important properties about the matrix, such as invertibility. For a 3x3 matrix, the determinant is calculated using a specific formula involving the elements of the matrix. A non-zero determinant indicates the system has a unique solution.

Systems of Linear Equations

A system of linear equations consists of multiple linear equations with the same set of variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. Methods include substitution, elimination, matrix operations, and Cramer's Rule, especially when dealing with three variables as in this problem.

In Exercises 37–44, use Cramer's Rule to solve each system. {x+y+z=02x−y+z=−1−x+3y−z=−8\(\begin{cases}\) x + y + z = 0 \\ 2x - y + z = -1 \\ -x + 3y - z = -8 \(\end{cases}\)⎩⎨⎧x+y+z=02x−y+z=−1−x+3y−z=−8