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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblem 1
Chapter 7, Problem 1

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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Identify the coefficients of each variable in the system of equations. For the first equation, the coefficients are 2 for x, 1 for y, and 2 for z.
For the second equation, the coefficients are 3 for x, -5 for y, and -1 for z (since it is '- z', the coefficient is -1).
For the third equation, the coefficients are 1 for x, -2 for y, and -3 for z.
Write these coefficients in a matrix form where each row corresponds to an equation and each column corresponds to a variable: x, y, and z respectively.
Add the constants from the right side of the equations as the last column to form the augmented matrix.

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Key Concepts

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

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 in algebra.
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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, facilitating systematic methods like row operations.
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Matrix Representation of Equations

Matrix representation translates linear equations into a compact form using rows and columns. This allows the use of matrix operations to solve systems efficiently, such as Gaussian elimination or matrix inversion, making it easier to handle multiple equations and variables.
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Related Practice
Textbook Question

Perform each matrix row operation and write the new matrix.

[122∣201−1∣2054∣1]−5R2+R3\(\begin{bmatrix}\)1 & 2 & 2 & \(\vert\) & 2 \\0 & 1 & -1 & \(\vert\) & 2 \\0 & 5 & 4 & \(\vert\) & 1\(\end{bmatrix}\)-5R_2 + R_3

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Textbook Question

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {5x+12y+z=102x+5y+2z=−1x+2y−3z=5\(\begin{cases}\)5x + 12y + z = 10 \\2x + 5y + 2z = -1 \(\x\) + 2y - 3z = 5\(\end{cases}\)

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Textbook Question

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

A=[4−3−54],B=[4354]A = \(\begin{bmatrix}\)4 & -3 \\-5 & 4\(\end{bmatrix}\), B = \(\begin{bmatrix}\)4 & 3 \\5 & 4\(\end{bmatrix}\)