In Exercises 37–44, use Cramer's Rule to solve each system. {x+2z=42y−z=52x+3y=13\begin{cases} x + 2z = 4 \\ 2y - z = 5 \\ 2x + 3y = 13 \end{cases}⎩⎨⎧x+2z=42y−z=52x+3y=13

Ch. 6 - Matrices and Determinants

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

문제 43

7장, 문제 43

In Exercises 37–44, use Cramer's Rule to solve each system. $\begin{cases}\) x + 2z = 4 \\ 2y - z = 5 \\ 2x + 3y = 13 \(\end{cases}$

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Write the system of equations in standard form, aligning variables x, y, and z: $\begin{cases} x + 0y + 2z = 4 \\ 0x + 2y - z = 5 \\ 2x + 3y + 0z = 13 \end{cases}$

Form the coefficient matrix $A$ from the coefficients of $x$, $y$, and $z$: $A = \begin{bmatrix} 1 & 0 & 2 \\ 0 & 2 & -1 \\ 2 & 3 & 0 \end{bmatrix}$

Calculate the determinant of matrix $A$, denoted as $\det(A)$, to ensure it is not zero (which would mean the system has a unique solution).

Form matrices $A_x$, $A_y$, and $A_z$ by replacing the respective columns of $A$ with the constants vector $\begin{bmatrix} 4 \\ 5 \\ 13 \end{bmatrix}$: - $A_x$ replaces the first column, - $A_y$ replaces the second column, - $A_z$ replaces the third column.

Calculate the determinants $\det(A_x)$, $\det(A_y)$, and $\det(A_z)$, then use Cramer's Rule to find the variables: $ x = \frac{\det(A_x)}{\det(A)}$, $ y = \frac{\det(A_y)}{\det(A)}$, $ z = \frac{\det(A_z)}{\det(A)}$.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 coefficient matrix has a non-zero determinant. 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 minors and cofactors. A non-zero determinant indicates the system has a unique solution.

Setting up the Coefficient Matrix and Constants Vector

To apply Cramer's Rule, the system of equations must be expressed in matrix form: Ax = b, where A is the coefficient matrix, x is the vector of variables, and b is the constants vector. Correctly identifying coefficients and constants from the equations is essential for forming these matrices accurately.

In Exercises 37–44, use Cramer's Rule to solve each system. {x+2z=42y−z=52x+3y=13\(\begin{cases}\) x + 2z = 4 \\ 2y - z = 5 \\ 2x + 3y = 13 \(\end{cases}\)⎩⎨⎧x+2z=42y−z=52x+3y=13