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

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

Problem 35

Chapter 7, Problem 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}$

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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 $x + y = 5$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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Related Practice

Textbook Question

Perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.

$A = [\begin{matrix} 4 & 0 \\ - 3 & 5 \\ 0 & 1 \end{matrix}], B = [\begin{matrix} 5 & 1 \\ - 2 & - 2 \end{matrix}], C = [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$

4B - 3C

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Textbook Question

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. $\begin{vmatrix}\)0.5 & 7 & 5 \\0.5 & 3 & 9 \\0.5 & 1 & 3\(\end{vmatrix}$

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Textbook Question

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases}\)2x + 2y + 7z = -1 \\2x + y + 2z = 2 \\4x + 6y + z = 15\(\end{cases}$

1145

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Textbook Question

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

$\begin{cases}\)w + x + y + z = 4 \\2w + x - 2y - z = 0 $\w$ - 2x - y - 2z = -2 \\3w + 2x + y + 3z = 4\(\end{cases}$

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Textbook Question

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 4 2 2 3 4 A = 6 1 B = 3 5 - 1 - 2 0

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Textbook Question

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 2 2 - 3 1 - 1 - 1 1 A = B = 1 1 - 2 1 5 4 10 5

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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

Verified video answer for a similar problem:

Key Concepts

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}$