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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 15
Chapter 7, Problem 15

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {2x+y−z=23x+3y−2z=3\(\begin{cases}\)2x + y - z = 2 \\3x + 3y - 2z = 3\(\end{cases}\)

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Write the system of equations as an augmented matrix: \[\left[\begin{array}{ccc|c} 2 & 1 & -1 & 2 \\ 3 & 3 & -2 & 3 \end{array}\right]\]
Use row operations to create a leading 1 in the first row, first column. For example, divide the first row by 2: \[R_1 \to \frac{1}{2} R_1\]
Eliminate the x-term in the second row by replacing the second row with \( R_2 - 3 \times R_1 \): \[R_2 \to R_2 - 3R_1\]
Simplify the second row to get a new equation involving only y and z. Then, if possible, create a leading 1 in the second row, second column by dividing the second row by the coefficient of y.
Use back substitution to express variables in terms of each other or constants, and write the solution set accordingly.

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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 two or more linear equations involving the same set of variables. The goal is to find values for the 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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Consistency and Types of Solutions

A system of linear equations can have a unique solution, infinitely many solutions, or no solution. Recognizing the system's consistency involves analyzing the row-echelon form for contradictions or free variables, which helps determine the nature of the solution set.
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Related Practice
Textbook Question

In Exercises 9 - 16, find the following matrices: c. - 4A

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Textbook Question

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

Textbook Question

For Exercises 11–22, use Cramer's Rule to solve each system. {4x−5y=172x+3y=3\(\begin{cases}\)4x - 5y = 17 \\2x + 3y = 3\(\end{cases}\)

Textbook Question

Use the fact that if A=[abcd]A = \(\begin{bmatrix}\) a & b \\ c & d \(\end{bmatrix}\), then A−1=1ad−bc[d−b−ca]A^{-1} = \(\frac{1}{ad-bc}\) \(\begin{bmatrix}\) d & -b \\ -c & a \(\end{bmatrix}\) to find the inverse of each matrix, if possible. Check that AA−1=I2AA^{-1} = I_2 and A−1A=I2A^{-1} A = I_2.

A=[3−1−42]A = \(\begin{bmatrix}\)3 & -1 \\-4 & 2\(\end{bmatrix}\)

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Textbook Question

Perform each matrix row operation and write the new matrix.

[1−32∣031−1∣72−21∣3]−3R1+R2\(\begin{bmatrix}\)1 & -3 & 2 & \(\vert\) & 0 \\3 & 1 & -1 & \(\vert\) & 7 \\2 & -2 & 1 & \(\vert\) & 3\(\end{bmatrix}\]\quad\) -3R_1 + R_2

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Textbook Question

In Exercises 9 - 16, find the following matrices: a. A + B