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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 3
Chapter 7, Problem 3

Write the augmented matrix for each system of linear equations.
{x−y+z=8y−12z=−15z=1\(\begin{cases}\)x - y + z = 8 \(\y\) - 12z = -15 \(\z\) = 1\(\end{cases}\)

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Identify the coefficients of each variable in the system of equations. For the first equation \(x - y + z = 8\), the coefficients are 1 for \(x\), -1 for \(y\), and 1 for \(z\).
For the second equation \(y - 12z = -15\), note that \(x\) is missing, so its coefficient is 0. The coefficients are 0 for \(x\), 1 for \(y\), and -12 for \(z\).
For the third equation \(z = 1\), both \(x\) and \(y\) are missing, so their coefficients are 0. The coefficients are 0 for \(x\), 0 for \(y\), and 1 for \(z\).
Write the augmented matrix by placing the coefficients of \(x\), \(y\), and \(z\) in the first three columns, and the constants on the right side of the equations in the last column.
The augmented matrix will have three rows (one for each equation) and four columns (three for variables and one for constants).

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Key Concepts

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

System 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 interpret and manipulate these systems is fundamental for forming matrices.
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Augmented Matrix

An augmented matrix represents a system of linear equations by combining the coefficient matrix and the constants into one matrix. Each row corresponds to an equation, and each column corresponds to a variable or the constants. This format simplifies solving systems using matrix operations.
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Matrix Notation and Construction

Matrix notation organizes coefficients and constants into rows and columns, making it easier to apply algebraic methods like row operations. Constructing the augmented matrix requires correctly placing coefficients of variables and constants in order, including zeros for missing variables.
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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

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

Evaluate each determinant in Exercises 1–10.

∣−4156∣\(\begin{vmatrix}\)-4 & 1 \\5 & 6\(\end{vmatrix}\)

Textbook Question

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

A=[−4013],B=[−2401]A = \(\begin{bmatrix}\) -4 & 0 \\ 1 & 3 \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) -2 & 4 \\ 0 & 1 \(\end{bmatrix}\)

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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