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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 3
Capitolo 7, Problema 3

Write the augmented matrix for each system of linear equations.
{x−y+z=8y−12z=−15z=1\(\begin{cases}\) x - y + z = 8 \\ y - 12z = -15 \\ z = 1 \(\end{cases}\)

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Identify the coefficients of each variable in the system of equations. For the first equation \(x - y + z = 8\), the coefficients are 1 for \(x\), -1 for \(y\), and 1 for \(z\).
For the second equation \(y - 12z = -15\), note that \(x\) is missing, so its coefficient is 0. The coefficients are 0 for \(x\), 1 for \(y\), and -12 for \(z\).
For the third equation \(z = 1\), both \(x\) and \(y\) are missing, so their coefficients are 0. The coefficients are 0 for \(x\), 0 for \(y\), and 1 for \(z\).
Write the augmented matrix by placing the coefficients of \(x\), \(y\), and \(z\) in the first three columns, and the constants on the right side of the equations in the last column.
The augmented matrix will have three rows (one for each equation) and four columns (three for variables and one for constants).

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System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to interpret and manipulate these systems is fundamental for forming matrices.
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Introduction to Systems of Linear Equations

Augmented Matrix

An augmented matrix represents a system of linear equations by combining the coefficient matrix and the constants into one matrix. Each row corresponds to an equation, and each column corresponds to a variable or the constants. This format simplifies solving systems using matrix operations.
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Introduction to Matrices

Matrix Notation and Construction

Matrix notation organizes coefficients and constants into rows and columns, making it easier to apply algebraic methods like row operations. Constructing the augmented matrix requires correctly placing coefficients of variables and constants in order, including zeros for missing variables.
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Interval Notation
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Domanda del libro di testo

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {5x+8y−6z=143x+4y−2z=8x+2y−2z=3\(\begin{cases}\) 5x + 8y - 6z = 14 \\ 3x + 4y - 2z = 8 \\ x + 2y - 2z = 3 \(\end{cases}\)

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Domanda del libro di testo

Write the augmented matrix for each system of linear equations.

{2x+y+2z=23x−5y−z=4x−2y−3z=−6\(\begin{cases}\) 2x + y + 2z = 2 \\ 3x - 5y - z = 4 \\ x - 2y - 3z = -6 \(\end{cases}\)

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a. Give the order of each matrix.


b. If A=[aij]A = [a_{ij}], identify a32a_{32} and a23a_{23}, or explain why identification is not possible.

[4−75−68−1]\(\begin{bmatrix}\) 4 & -7 & 5 \\ -6 & 8 & -1 \(\end{bmatrix}\)

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Evaluate each determinant in Exercises 1–10.

∣−4156∣\(\begin{vmatrix}\) -4 & 1 \\ 5 & 6 \(\end{vmatrix}\)

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Find the products AB and BA to determine whether B is the multiplicative inverse of A.

A=[−4013],B=[−2401]A = \(\begin{bmatrix}\) -4 & 0 \\ 1 & 3 \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) -2 & 4 \\ 0 & 1 \(\end{bmatrix}\)

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Domanda del libro di testo

a. Give the order of each matrix.


b. If A = [aᵢⱼ] , identify a₃₂ and a₂₃, or explain why identification is not possible.

[1−5πe07−6−π−21211−15]\(\begin{bmatrix}\) 1 & -5 & \(\pi\) & e \\ 0 & 7 & -6 & -\(\pi\) \\ -2 & \(\frac{1}{2}\) & 11 & -\(\frac{1}{5}\) \(\end{bmatrix}\)

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