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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblema 9
Capitolo 7, Problema 9

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {w−2x−y−3z=−9w+x−y=03w+4x+z=62x−2y+z=3\(\begin{cases}\) w - 2x - y - 3z = -9 \\ w + x - y = 0 \\ 3w + 4x + z = 6 \\ 2x - 2y + z = 3 \(\end{cases}\)

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Write the system of equations as an augmented matrix. The system is: \(\begin{cases} w - 2x - y - 3z = -9 \\ w + x - y + 0z = 0 \\ 3w + 4x + 0y + z = 6 \\ 0w + 2x - 2y + z = 3 \end{cases}\) The augmented matrix is: \(\left[\begin{array}{cccc|c} 1 & -2 & -1 & -3 & -9 \\ 1 & 1 & -1 & 0 & 0 \\ 3 & 4 & 0 & 1 & 6 \\ 0 & 2 & -2 & 1 & 3 \end{array}\right]\)
Use row operations to create zeros below the leading 1 in the first column (pivot position). For example, subtract Row 1 from Row 2 to eliminate the first element in Row 2, and subtract 3 times Row 1 from Row 3 to eliminate the first element in Row 3.
Next, move to the second column and create a leading 1 in the second row, second column if necessary. Then use row operations to create zeros above and below this pivot. This involves manipulating Rows 3 and 4 to eliminate the second column entries below and above the pivot.
Continue this process for the third and fourth columns, creating leading 1s (pivots) and zeros in all other positions of those columns. This will transform the matrix into reduced row echelon form (RREF).
Once the matrix is in RREF, translate the matrix back into equations. From these equations, express the variables in terms of any free variables if they exist, or find the unique solution if the system is consistent and determined.

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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 such 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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Solving Systems of Equations - Elimination

Row Operations and Consistency of Systems

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 solutions) or inconsistent (no solutions). Recognizing inconsistent rows is key to concluding no solution exists.
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Performing Row Operations on Matrices
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A=[123134143],B=[72−312−12012−121−12]A = \(\begin{bmatrix}\) 1 & 2 & 3 \\ 1 & 3 & 4 \\ 1 & 4 & 3 \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) \(\frac{7}{2}\) & -3 & \(\frac{1}{2}\) \\ -\(\frac{1}{2}\) & 0 & \(\frac{1}{2}\) \\ -\(\frac{1}{2}\) & 1 & -\(\frac{1}{2}\) \(\end{bmatrix}\)

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{2w+5x−3y+z=23x+y=4w−x+5y=95w−5x−2y=1\(\begin{cases}\) 2w + 5x - 3y + z = 2 \\ 3x + y = 4 \\ w - x + 5y = 9 \\ 5w - 5x - 2y = 1 \(\end{cases}\)

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