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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblema 35
Capitolo 7, Problema 35

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 \(\end{bmatrix}\)

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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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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.
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Finding Zeros & Their Multiplicity

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.
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Introduction to Systems of Linear Equations

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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Equations with Two Variables
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Domanda del libro di testo

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=[40−3501],B=[51−2−2],C=[1−1−11]A=\(\begin{bmatrix}\)4 & 0\\ -3 & 5\\ 0 & 1\(\end{bmatrix}\),B=\(\begin{bmatrix}\)5 & 1\\ -2 & -2\(\end{bmatrix}\),C=\(\begin{bmatrix}\)1 & -1\\ -1 & 1\(\end{bmatrix}\)

4B - 3C

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Domanda del libro di testo

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. ∣0.5750.5390.513∣\(\begin{vmatrix}\) 0.5 & 7 & 5 \\ 0.5 & 3 & 9 \\ 0.5 & 1 & 3 \(\end{vmatrix}\)

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Domanda del libro di testo

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

{2x+2y+7z=−12x+y+2z=24x+6y+z=15\(\begin{cases}\) 2x + 2y + 7z = -1 \\ 2x + y + 2z = 2 \\ 4x + 6y + z = 15 \(\end{cases}\)

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Domanda del libro di testo

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

{w+x+y+z=42w+x−2y−z=0w−2x−y−2z=−23w+2x+y+3z=4\(\begin{cases}\) 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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Domanda del libro di testo

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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Domanda del libro di testo

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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