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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 37
Chapter 7, Problem 37

In Exercises 37–44, use Cramer's Rule to solve each system. {x+y+z=02x−y+z=−1−x+3y−z=−8\(\begin{cases}\)x + y + z = 0 \\2x - y + z = -1 \\-x + 3y - z = -8\(\end{cases}\)

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Write the system of equations in matrix form as \(A\mathbf{x} = \mathbf{b}\), where \(A\) is the coefficient matrix, \(\mathbf{x}\) is the column vector of variables, and \(\mathbf{b}\) is the constants vector. For this system, \(A = \begin{bmatrix} 1 & 1 & 1 \\ 2 & -1 & 1 \\ -1 & 3 & -1 \end{bmatrix}\), \(\mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}\), and \(\mathbf{b} = \begin{bmatrix} 0 \\ -1 \\ -8 \end{bmatrix}\).
Calculate the determinant of the coefficient matrix \(A\), denoted as \(\det(A)\). This determinant must be non-zero to apply Cramer's Rule.
Form three new matrices \(A_x\), \(A_y\), and \(A_z\) by replacing the respective columns of \(A\) with the constants vector \(\mathbf{b}\). Specifically, \(A_x\) replaces the first column, \(A_y\) replaces the second column, and \(A_z\) replaces the third column with \(\mathbf{b}\).
Calculate the determinants of these new matrices: \(\det(A_x)\), \(\det(A_y)\), and \(\det(A_z)\).
Use Cramer's Rule to find the solutions for \(x\), \(y\), and \(z\) by dividing each determinant by \(\det(A)\): \(x = \frac{\det(A_x)}{\det(A)}\), \(y = \frac{\det(A_y)}{\det(A)}\), and \(z = \frac{\det(A_z)}{\det(A)}\).

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Key Concepts

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

Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. It applies when the system has the same number of equations as unknowns and the determinant of the coefficient matrix is non-zero. Each variable is found by replacing the corresponding column in the coefficient matrix with the constants vector and calculating the determinant ratio.
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Determinants of Matrices

The determinant is a scalar value that can be computed from a square matrix and provides important properties about the matrix, such as invertibility. For a 3x3 matrix, the determinant is calculated using a specific formula involving the elements of the matrix. A non-zero determinant indicates the system has a unique solution.
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Systems of Linear Equations

A system of linear equations consists of multiple linear equations with the same set of variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. Methods include substitution, elimination, matrix operations, and Cramer's Rule, especially when dealing with three variables as in this problem.
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Related Practice
Textbook Question

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

A=[40−3501],B=[51−2−2],C=[1−1−11]A=\(\begin{bmatrix}\)4 & 0\\ -3 & 5\\ 0 & 1\(\end{bmatrix}\),B=\(\begin{bmatrix}\)5 & 1\\ -2 & -2\(\end{bmatrix}\),C=\(\begin{bmatrix}\)1 & -1\\ -1 & 1\(\end{bmatrix}\)

4B - 3C

Textbook Question

a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.

{2x+6y+6z=82x+7y+6z=102x+7y+7z=9The inverse of [266276277] is [720−3−1000−11].\(\begin{cases}\)2x + 6y + 6z = 8 \\2x + 7y + 6z = 10 \\2x + 7y + 7z = 9\(\end{cases}\[\text{The inverse of }\]\begin{bmatrix}\)2 & 6 & 6 \\2 & 7 & 6 \\2 & 7 & 7\(\end{bmatrix}\[\text{ is }\]\begin{bmatrix}\[\frac{7}{2}\) & 0 & -3 \\-1 & 0 & 0 \\0 & -1 & 1\(\end{bmatrix}\]\text{.}\)

Textbook Question

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. ∣0.5750.5390.513∣\(\begin{vmatrix}\)0.5 & 7 & 5 \\0.5 & 3 & 9 \\0.5 & 1 & 3\(\end{vmatrix}\)

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Textbook Question

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

{w+x+y+z=42w+x−2y−z=0w−2x−y−2z=−23w+2x+y+3z=4\(\begin{cases}\)w + x + y + z = 4 \\2w + x - 2y - z = 0 \(\w\) - 2x - y - 2z = -2 \\3w + 2x + y + 3z = 4\(\end{cases}\)

Textbook Question

In Exercises 37–38, find the products and to determine whether B is the multiplicative inverse of A.

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Textbook Question

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 2 2 - 3 1 - 1 - 1 1 A = B = 1 1 - 2 1 5 4 10 5