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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 9
Chapter 7, Problem 9

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

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Write the given system of linear equations in augmented matrix form. The augmented matrix combines the coefficients of the variables and the constants from the equations into a single matrix.
Use row operations (row swapping, scaling a row by a nonzero constant, or adding/subtracting multiples of one row to another) to transform the augmented matrix into row-echelon form. The goal is to create zeros below the pivot positions (leading entries in each row).
Continue applying row operations to transform the matrix into reduced row-echelon form (RREF). In RREF, each pivot is 1, and all other entries in the pivot's column are 0.
Interpret the resulting matrix. If the system has a unique solution, the RREF will clearly show the values of the variables. If there are free variables (columns without pivots), express the solution in terms of these free variables. If a row in the matrix corresponds to an inconsistent equation (e.g., 0 = 1), conclude that no solution exists.
Write the complete solution, if it exists, in terms of the variables. If there are free variables, express the solution as a parametric equation. If no solution exists, state that the system is inconsistent.

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Key Concepts

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

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations. It involves transforming the system's augmented matrix into row echelon form using a series of row operations, which include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting rows. This process simplifies the equations, making it easier to find solutions or determine if no solution exists.
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Row Echelon Form

Row echelon form is a specific arrangement of a matrix where all non-zero rows are above any rows of all zeros, and the leading coefficient of each non-zero row (the first non-zero number from the left) is to the right of the leading coefficient of the previous row. This form is crucial in Gaussian elimination as it allows for back substitution to find the values of the variables in a linear system.
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Existence of Solutions

In the context of linear systems, the existence of solutions refers to whether a system has one unique solution, infinitely many solutions, or no solution at all. This can be determined during the Gaussian elimination process by examining the final row echelon form of the augmented matrix. If a row leads to a contradiction (like 0 = 1), the system has no solution; if there are free variables, it indicates infinitely many solutions.
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Related Practice
Textbook Question

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

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 & -\(\frac{1}{2}\) \(\end{bmatrix}\)

Textbook Question

Evaluate each determinant in Exercises 1–10.∣121218−34∣\(\begin{vmatrix}\[\frac{1}{2}\) & \(\frac{1}{2}\) \(\frac{1}{8}\) & - \(\frac{3}{4}\]\end{vmatrix}\)

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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.

[503∣−1101−4∣12720∣3]\(\begin{bmatrix}\)5 & 0 & 3 & \(\vert\) & -11 \\0 & 1 & -4 & \(\vert\) & 12 \\7 & 2 & 0 & \(\vert\) & 3\(\end{bmatrix}\)

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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 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {8x+5y+11z=30−x−4y+2z=32x−y+5z=12\(\begin{cases}\)8x + 5y + 11z = 30 \\-x - 4y + 2z = 3 \\2x - y + 5z = 12\(\end{cases}\)

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Textbook Question

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

A=[010001100],B=[001100010]A = \(\begin{bmatrix}\) 0 & 1 & 0 \\ 0 & 0 & 1 \\ 1 & 0 & 0 \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) 0 & 0 & 1 \\ 1 & 0 & 0 \\ 0 & 1 & 0 \(\end{bmatrix}\)