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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

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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Write the system of equations as an augmented matrix. Each row corresponds to an equation, and each column corresponds to the coefficients of variables w, x, y, z, and the constants on the right side: \[\begin{bmatrix} 2 & 1 & -1 & 0 & | & 3 \\ 1 & -3 & 2 & 0 & | & -4 \\ 3 & 1 & -3 & 1 & | & 1 \\ 1 & 2 & -4 & -1 & | & -2 \end{bmatrix}\]
Use row operations to create zeros below the leading 1 in the first column (pivot position). For example, use the second row (which has a 1 in the first column) to eliminate the w-terms in the first, third, and fourth rows by appropriate row replacements.
Move to the second column and create a leading 1 in the second row, second column position if necessary. Then use this pivot to eliminate the x-terms in the rows below it by row operations, creating zeros below the pivot.
Continue this process for the third and fourth columns, creating leading 1s (pivots) and zeros below them, transforming the matrix into an upper triangular (row echelon) form.
Once in row echelon form, use back substitution to solve for the variables starting from the bottom row up, expressing each variable in terms of constants or previously found variables to find the complete solution.

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Key Concepts

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

Systems 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 these variables that satisfy all equations simultaneously. Understanding how to represent and interpret these systems is fundamental before applying solution methods.
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Introduction to Systems of Linear Equations

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations by transforming the system's augmented matrix into row-echelon form using row operations. This process simplifies the system, making it easier to solve through back substitution or to determine if no solution exists.
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Row Operations and Row-Echelon Form

Row operations include swapping rows, multiplying a row by a nonzero scalar, and adding multiples of one row to another. These operations are used to convert the augmented matrix into row-echelon form, where the matrix has a stair-step pattern of leading coefficients, facilitating the solution of the system.
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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: c. - 4A

Textbook Question

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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Textbook Question

In Exercises 9 - 16, find the following matrices: d. - 3A + 2B

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