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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 5
Chapter 7, Problem 5

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {3x+4y+2z=34x−2y−8z=−4x+y−z=3\(\begin{cases}\)3x + 4y + 2z = 3 \\4x - 2y - 8z = -4 \(\x\) + y - z = 3\(\end{cases}\)

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1
Write the system of equations as an augmented matrix: \[\left[\begin{array}{ccc|c} 3 & 4 & 2 & 3 \\ 4 & -2 & -8 & -4 \\ 1 & 1 & -1 & 3 \end{array}\right]\]
Use row operations to create a leading 1 in the first row, first column. For example, swap rows if needed or divide the first row by the coefficient of \(x\) in the first equation.
Eliminate the \(x\)-terms from the second and third rows by replacing those rows with suitable linear combinations of the first row and themselves, aiming to get zeros below the leading 1 in the first column.
Move to the second row and create a leading 1 in the second column (pivot position). Then eliminate the \(y\)-term from the third row by using row operations.
Finally, use back substitution starting from the last row to express \(z\), then substitute back to find \(y\), and then \(x\), thus finding the complete solution to the system.

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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 by back substitution or to determine if no solution exists.
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Row Operations and Consistency

Row operations include swapping rows, multiplying a row by a nonzero scalar, and adding multiples of one row to another. These operations preserve the solution set and help identify if the system is consistent (has at least one solution) or inconsistent (no solution). Recognizing inconsistent rows is key to concluding no solution exists.
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Related Practice
Textbook Question

In Exercises 3–5, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

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Textbook Question

Evaluate each determinant in Exercises 1–10.

∣−7142−4∣\(\begin{vmatrix}\)-7 & 14 \\2 & -4\(\end{vmatrix}\)

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Textbook Question

In Exercises 5 - 8, find values for the variables so that the matrices in each exercise are equal. [x4]=[6y]\(\begin{bmatrix}\)x \\4\(\end{bmatrix}\)=\(\begin{bmatrix}\)6 \(\y\]\end{bmatrix}\)

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Textbook Question

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

{3x1+5x2−8x3+5x4=−8x1+2x2−3x3+x4=−72x1+3x2−7x3+3x4=−114x1+8x2−10x3+7x4=−10\(\begin{cases}\) 3x_1 + 5x_2 - 8x_3 + 5x_4 = -8 \\ x_1 + 2x_2 - 3x_3 + x_4 = -7 \\ 2x_1 + 3x_2 - 7x_3 + 3x_4 = -11 \\ 4x_1 + 8x_2 - 10x_3 + 7x_4 = -10 \(\end{cases}\)

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Textbook Question

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

{x+2y+3z=−52x+y+z=1x+y−z=8\(\begin{cases}\) x + 2y + 3z = -5 \\ 2x + y + z = 1 \\ x + y - z = 8 \(\end{cases}\)

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Textbook Question

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

A=[−2132−12],B=[1234]A = \(\begin{bmatrix}\) -2 & 1 \\ \(\frac{3}{2}\) & -\(\frac{1}{2}\) \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) 1 & 2 \\ 3 & 4 \(\end{bmatrix}\)

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