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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 19
Chapter 7, Problem 19

For Exercises 11–22, use Cramer's Rule to solve each system. {3x−4y=42x+2y=12\(\begin{cases}\)3x - 4y = 4 \\2x + 2y = 12\(\end{cases}\)

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Write the system of equations in standard form: \(\begin{cases} 3x - 4y = 4 \\ 2x + 2y = 12 \end{cases}\)
Identify the coefficients for the variables and constants: \(A = \begin{bmatrix} 3 & -4 \\ 2 & 2 \end{bmatrix}\), \(\mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix}\), \(\mathbf{b} = \begin{bmatrix} 4 \\ 12 \end{bmatrix}\)
Calculate the determinant of matrix \(A\), denoted as \(D\): \(D = \det(A) = (3)(2) - (-4)(2)\)
Form matrices \(A_x\) and \(A_y\) by replacing the respective columns of \(A\) with the constants vector \(\mathbf{b}\): \(A_x = \begin{bmatrix} 4 & -4 \\ 12 & 2 \end{bmatrix}\), \(A_y = \begin{bmatrix} 3 & 4 \\ 2 & 12 \end{bmatrix}\)
Calculate the determinants \(D_x = \det(A_x)\) and \(D_y = \det(A_y)\), then use Cramer's Rule to find the solutions: \(x = \frac{D_x}{D}\) and \(y = \frac{D_y}{D}\)

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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 to systems with the same number of equations and unknowns, where the solution for each variable is found by replacing the corresponding column of the coefficient matrix with the constants vector and dividing by the determinant of the coefficient matrix.
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Determinants of 2x2 Matrices

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This value is crucial in Cramer's Rule, as it determines whether the system has a unique solution (non-zero determinant) or not. Calculating determinants accurately is essential for applying Cramer's Rule.
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Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. Understanding how to represent and manipulate these systems, such as writing them in matrix form, is fundamental for applying methods like Cramer's Rule to find solutions.
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Related Practice
Textbook Question

In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. -5(A+D)

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Textbook Question

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {x+y−2z=23x−y−6z=−7\(\begin{cases}\)x + y - 2z = 2 \\3x - y - 6z = -7\(\end{cases}\)

Textbook Question

In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. 3A+2D

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Textbook Question

Let A=[−3−72−950]A = \(\begin{bmatrix}\)-3 & -7 \\2 & -9 \\5 & 0\(\end{bmatrix}\) and B=[−5−1003−4]B = \(\begin{bmatrix}\)-5 & -1 \\0 & 0 \\3 & -4\(\end{bmatrix}\). Solve each matrix equation for X. 2X + A = B

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Textbook Question

In Exercises 19–20, a few steps in the process of simplifying the given matrix to row-echelon form, with 1s down the diagonal from upper left to lower right, and 0s below the 1s, are shown. Fill in the missing numbers in the steps that are shown.

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Textbook Question

Perform each matrix row operation and write the new matrix.

[1−111∣301−2−1∣02034∣115124∣6]−2R1+R3−5R1+R4\(\begin{bmatrix}\)1 & -1 & 1 & 1 & \(\vert\) & 3 \\0 & 1 & -2 & -1 & \(\vert\) & 0 \\2 & 0 & 3 & 4 & \(\vert\) & 11 \\5 & 1 & 2 & 4 & \(\vert\) & 6\(\end{bmatrix}\[\quad\]\begin{array}{l}\)-2R_1 + R_3 \\-5R_1 + R_4\(\end{array}\)

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