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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 11
Chapter 7, Problem 11

For Exercises 11–22, use Cramer's Rule to solve each system. {x+y=7x−y=3\(\begin{cases}\)x + y = 7 \(\x\) - y = 3\(\end{cases}\)

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Write the system of equations in standard form: \\ \(x + y = 7\) and \(x - y = 3\).
Identify the coefficient matrix \(A\), the variable matrix \(X\), and the constant matrix \(B\(: \\ \)A = \begin{bmatrix} 1 & 1 \\ 1 & -1 \end{bmatrix}\), \(X = \begin{bmatrix} x \\ y \end{bmatrix}\), \(B = \begin{bmatrix} 7 \\ 3 \end{bmatrix}\).
Calculate the determinant of the coefficient matrix \(A\(: \\ \)D = \det(A) = (1)(-1) - (1)(1)\).
Form matrices \(A_x\) and \(A_y\) by replacing the respective columns of \(A\) with the constant matrix \(B\(: \\ \)A_x = \begin{bmatrix} 7 & 1 \\ 3 & -1 \end{bmatrix}\) and \(A_y = \begin{bmatrix} 1 & 7 \\ 1 & 3 \end{bmatrix}\).
Calculate the determinants \(D_x = \det(A_x)\) and \(D_y = \det(A_y)\), then use Cramer's Rule to find the variables: \\ \(x = \frac{D_x}{D}\) 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 square systems where the number of equations equals the number of variables. The solution for each variable is found by replacing the corresponding column of the coefficient matrix with the constants vector and dividing the determinant of this new matrix by the determinant of the coefficient matrix.
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Cramer's Rule - 2 Equations with 2 Unknowns

Determinants of 2x2 Matrices

The determinant of a 2x2 matrix [[a, b], [c, d]] is calculated as ad - bc. This scalar value helps determine if the system has a unique solution (non-zero determinant) or not. In Cramer's Rule, determinants are used to find the values of variables by comparing the determinant of the coefficient matrix and modified matrices.
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Systems of Linear Equations

A system of linear equations consists of two or more linear equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. Understanding how to represent and manipulate these systems is essential for applying methods like Cramer's Rule.
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Related Practice
Textbook Question

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

A=[00−21−101101−10100−1],B=[1203011101011202]A = \(\begin{bmatrix}\) 0 & 0 & -2 & 1 \\ -1 & 0 & 1 & 1 \\ 0 & 1 & -1 & 0 \\ 1 & 0 & 0 & -1 \(\end{bmatrix}\), \(\quad\) B = \(\begin{bmatrix}\) 1 & 2 & 0 & 3 \\ 0 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 1 & 2 & 0 & 2 \(\end{bmatrix}\)

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Textbook Question

In Exercises 8–11, use Gaussian elimination to find the complete solution to each system, or show that none exists.

Textbook Question

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

Textbook Question

Write the system of linear equations represented by the augmented matrix. Use x, y, and z, or, if necessary, w, x, y, and z, for the variables.

[1141∣3−11−10∣72005∣1100124∣5]\(\begin{bmatrix}\)1 & 1 & 4 & 1 & \(\vert\) & 3 \\-1 & 1 & -1 & 0 & \(\vert\) & 7 \\2 & 0 & 0 & 5 & \(\vert\) & 11 \\0 & 0 & 12 & 4 & \(\vert\) & 5\(\end{bmatrix}\)

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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. {2w+x−y=3w−3x+2y=−43w+x−3y+z=1w+2x−4y−z=−2\(\begin{cases}\)2w + x - y = 3 \(\w\) - 3x + 2y = -4 \\3w + x - 3y + z = 1 \(\w\) + 2x - 4y - z = -2\(\end{cases}\)

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Textbook Question

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

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