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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 23
Chapter 7, Problem 23

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {w+2x+3y−z=72x−3y+z=4w−4x+y=3\(\begin{cases}\)w + 2x + 3y - z = 7 \\2x - 3y + z = 4 \(\w\) - 4x + y = 3\(\end{cases}\)

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Write the system of equations as an augmented matrix. The system is: \(\begin{cases} w + 2x + 3y - z = 7 \\ 0w + 2x - 3y + z = 4 \\ w - 4x + y + 0z = 3 \end{cases}\) So the augmented matrix is: \(\left[ \begin{array}{cccc|c} 1 & 2 & 3 & -1 & 7 \\ 0 & 2 & -3 & 1 & 4 \\ 1 & -4 & 1 & 0 & 3 \end{array} \right]\)
Use row operations to create zeros below the leading 1 in the first column. Specifically, subtract Row 1 from Row 3 to eliminate the \(w\) term in Row 3: \(R_3 \leftarrow R_3 - R_1\)
Next, focus on the second column. Use the second row to create a leading 1 if necessary, and then eliminate the \(x\) term in the third row by appropriate row operations. This will help in forming an upper triangular matrix.
Continue applying Gaussian elimination steps to get the matrix into row echelon form, where each leading coefficient is 1 and all entries below each leading 1 are zero. This may involve scaling rows and adding multiples of one row to another.
Once in row echelon form, use back substitution to express the variables \(w\), \(x\), \(y\), and \(z\) in terms of constants or parameters if there are free variables, 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 through back substitution or to determine if no solution exists.
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Consistency and Solution Types of Systems

A system of linear equations can be consistent (having at least one solution) or inconsistent (no solution). Consistent systems may have a unique solution or infinitely many solutions. Recognizing these outcomes during Gaussian elimination helps interpret the results correctly.
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Related Practice
Textbook Question

Let A=[−3−72−950]A = \(\begin{bmatrix}\)-3 & -7 \\2 & -9 \\5 & 0\(\end{bmatrix}\) and B=[−5−1003−4]B = \(\begin{bmatrix}\)-5 & -1 \\0 & 0 \\3 & -4\(\end{bmatrix}\). Solve each matrix equation for X. 4A + 3B = - 2X

Textbook Question

Perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. BD

A=[2−1253−1]B=[0−2321−5]C=[123−112−121]D=[−2313−24]A=\(\begin{bmatrix}\)2 & -1 & 2\\ 5 & 3 & -1\(\end{bmatrix}\[\quad\) B=\(\begin{bmatrix}\)0 & -2\\ 3 & 2\\ 1 & -5\(\end{bmatrix}\)C=\(\begin{bmatrix}\)1 & 2 & 3\\ -1 & 1 & 2\\ -1 & 2 & 1\(\end{bmatrix}\]\quad\) D=\(\begin{bmatrix}\)-2 & 3 & 1\\ 3 & -2 & 4\(\end{bmatrix}\)

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Textbook Question

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

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

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Textbook Question

In Exercises 23–30, use expansion by minors to evaluate each determinant. ∣310−340−13−5∣\(\begin{vmatrix}\)3 & 1 & 0 \\-3 & 4 & 0 \\-1 & 3 & -5\(\end{vmatrix}\)

Textbook Question

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

{x+3y=0x+y+z=13x−y−z=11\(\begin{cases}\)x + 3y = 0 \(\x\) + y + z = 1 \\3x - y - z = 11\(\end{cases}\)

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Textbook Question

Let A=[−3−72−950]A = \(\begin{bmatrix}\)-3 & -7 \\2 & -9 \\5 & 0\(\end{bmatrix}\) and B=[−5−1003−4]B = \(\begin{bmatrix}\)-5 & -1 \\0 & 0 \\3 & -4\(\end{bmatrix}\). Solve each matrix equation for X. B - X = 4A