Write each matrix equation as a system of linear equations without matrices.[20−1030110][xyz]=[695]\begin{bmatrix} 2 & 0 & -1 \\ 0 & 3 & 0 \\ 1 & 1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 9 \\ 5 \end{bmatrix}201031−100xyz=695

Ch. 6 - Matrices and Determinants

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

문제 35

7장, 문제 35

Write each matrix equation as a system of linear equations without matrices.
$\begin{bmatrix}\) 2 & 0 & -1 \\ 0 & 3 & 0 \\ 1 & 1 & 0 $\end{bmatrix}$ $\begin{bmatrix}$ x \\ y \\ z $\end{bmatrix}$ = $\begin{bmatrix}$ 6 \\ 9 \\ 5 \(\end{bmatrix}$

검증된 단계별 안내

Identify the matrix equation given: $\begin{bmatrix} 2 & 0 & -1 \\ 0 & 3 & 0 \\ 1 & 1 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 9 \\ 5 \end{bmatrix}$.

Recall that multiplying a matrix by a vector corresponds to taking the dot product of each row of the matrix with the vector. This results in a system of linear equations.

Write the first row multiplied by the vector: $2x + 0 \cdot y + (-1)z = 6$, which simplifies to $2x - z = 6$.

Write the second row multiplied by the vector: $0 \cdot x + 3y + 0 \cdot z = 9$, which simplifies to $3y = 9$.

Write the third row multiplied by the vector: $1x + 1y + 0 \cdot z = 5$, which simplifies to $x + y = 5$.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

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

Matrix Multiplication

Matrix multiplication involves multiplying each row of the first matrix by each column of the second matrix and summing the products. In this problem, multiplying the coefficient matrix by the variable matrix results in a new matrix representing the system's left-hand side expressions.

System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same variables. Writing the matrix equation as a system means expressing each row multiplication as an individual linear equation equated to the corresponding constant.

Variables and Constants in Matrix Form

The variables (x, y, z) are represented as a column matrix, and the constants on the right side form another column matrix. Understanding how these correspond to the coefficients and constants in the system is essential for translating between matrix and equation forms.

Write each matrix equation as a system of linear equations without matrices.[20−1030110][xyz]=[695]\(\begin{bmatrix}\) 2 & 0 & -1 \\ 0 & 3 & 0 \\ 1 & 1 & 0 \(\end{bmatrix}\) \(\begin{bmatrix}\) x \\ y \\ z \(\end{bmatrix}\) = \(\begin{bmatrix}\) 6 \\ 9 \\ 5 \(\end{bmatrix}\)201031−100xyz=695

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Matrix Multiplication

System of Linear Equations

Variables and Constants in Matrix Form

Write each matrix equation as a system of linear equations without matrices.
$\begin{bmatrix}\) 2 & 0 & -1 \\ 0 & 3 & 0 \\ 1 & 1 & 0 \(\end{bmatrix}\) \(\begin{bmatrix}\) x \\ y \\ z \(\end{bmatrix}\) = \(\begin{bmatrix}\) 6 \\ 9 \\ 5 \(\end{bmatrix}$