Ch. 6 - Matrices and Determinants

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

Problem 45

Chapter 7, Problem 45

In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussian elimination to solve the system.

Verified step by step guidance

Step 1: Write the system of equations in matrix form as A * X = B, where A is the coefficient matrix, X is the variable vector, and B is the constants vector. Here, A = [[2, -3, 2], [2, 3, -2], [2, -9, 6]], X = [x, y, z]^T, and B = [4, 6, 2]^T.

Step 2: Calculate the determinant of the coefficient matrix A, denoted as det(A). If det(A) = 0, Cramer's Rule cannot be used because it requires a non-zero determinant to find a unique solution.

Step 3: Since det(A) = 0 (as you will find upon calculation), explain that the system either has infinitely many solutions or no solution, so Cramer's Rule is not applicable.

Step 4: Use Gaussian elimination to solve the system. Start by writing the augmented matrix [A | B]: [[2, -3, 2, 4], [2, 3, -2, 6], [2, -9, 6, 2]].

Step 5: Perform row operations to reduce the augmented matrix to row-echelon form, then use back substitution to find the values of x, y, and z or determine if the system is inconsistent or dependent.

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Key Concepts

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

Cramer's Rule and Determinants

Cramer's Rule solves a system of linear equations using determinants, applicable only when the coefficient matrix has a nonzero determinant. If the determinant is zero, the system is either dependent or inconsistent, making Cramer's Rule unusable.

Gaussian Elimination

Gaussian elimination is a method to solve systems of linear equations by transforming the augmented matrix into row-echelon form using row operations. This process simplifies the system, allowing back-substitution to find the solution or determine if none exists.

Consistency and Dependence of Systems

A system is consistent if it has at least one solution and inconsistent if it has none. Dependence occurs when equations are multiples or linear combinations of others, leading to infinite solutions or no unique solution, which affects the applicability of solution methods.

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Related Practice

Textbook Question

In Exercises 37 - 44, perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.

$A = [\begin{matrix} 4 & 0 \\ - 3 & 5 \\ 0 & 1 \end{matrix}], B = [\begin{matrix} 5 & 1 \\ - 2 & - 2 \end{matrix}], C = [\begin{matrix} 1 & - 1 \\ - 1 & 1 \end{matrix}]$

A(BC)

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Textbook Question

In Exercises 46–51, evaluate each determinant.

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Textbook Question

In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussian elimination to solve the system.

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Textbook Question

In Exercises 43–44, (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.

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Textbook Question

In Exercises 37–44, use Cramer's Rule to solve each system. $\begin{cases}\)x + 2z = 4 \\2y - z = 5 \\2x + 3y = 13\(\end{cases}$

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Textbook Question

Evaluate each determinant.

$∣ \begin{matrix} 4 & 7 & 0 \\ - 5 & 6 & 0 \\ 3 & 2 & - 4 \end{matrix} ∣$

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