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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 11
Chapter 7, Problem 11

Write the system of linear equations represented by the augmented matrix. Use x, y, and z, or, if necessary, w, x, y, and z, for the variables.
[1141∣3−11−10∣72005∣1100124∣5]\(\begin{bmatrix}\)1 & 1 & 4 & 1 & \(\vert\) & 3 \\-1 & 1 & -1 & 0 & \(\vert\) & 7 \\2 & 0 & 0 & 5 & \(\vert\) & 11 \\0 & 0 & 12 & 4 & \(\vert\) & 5\(\end{bmatrix}\)

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Identify the variables corresponding to each column in the augmented matrix. Since there are four columns before the augmented part, use variables w, x, y, and z in that order.
Write the first row as an equation by multiplying each variable by the corresponding coefficient and setting it equal to the augmented value: \(1w + 1x + 4y + 1z = 3\).
Write the second row as an equation: \(-1w + 1x - 1y + 0z = 7\).
Write the third row as an equation: \(0w + 2x + 0y + 5z = 11\).
Write the fourth row as an equation: \(0w + 0x + 12y + 4z = 5\).

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Key Concepts

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

Augmented Matrix Representation

An augmented matrix combines the coefficients of variables and constants from a system of linear equations into a single 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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System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Writing the system from an augmented matrix involves translating each row into an equation.
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Variables and Their Correspondence

Variables in the system correspond to the columns of the augmented matrix, excluding the last column which represents constants. In this problem, the variables are w, x, y, and z, matching the four columns of coefficients. Correctly identifying variables is essential to write the system accurately.
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Related Practice
Textbook Question

For Exercises 11–22, use Cramer's Rule to solve each system. {x+y=7x−y=3\(\begin{cases}\)x + y = 7 \(\x\) - y = 3\(\end{cases}\)

Textbook Question

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

A=[00−21−101101−10100−1],B=[1203011101011202]A = \(\begin{bmatrix}\) 0 & 0 & -2 & 1 \\ -1 & 0 & 1 & 1 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & -1 \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) 1 & 2 & 0 & 3 \\ 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 1 & 2 & 0 & 2 \(\end{bmatrix}\)

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Textbook Question

In Exercises 8–11, use Gaussian elimination to find the complete solution to each system, or show that none exists.

Textbook Question

In Exercises 9 - 16, find the following matrices: a. A + B

Textbook Question

In Exercises 9 - 16, find the following matrices: b. A - B

Textbook Question

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {2w+x−y=3w−3x+2y=−43w+x−3y+z=1w+2x−4y−z=−2\(\begin{cases}\)2w + x - y = 3 \(\w\) - 3x + 2y = -4 \\3w + x - 3y + z = 1 \(\w\) + 2x - 4y - z = -2\(\end{cases}\)

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