Skip to main content
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 41
Chapter 7, Problem 41

In Exercises 37–44, use Cramer's Rule to solve each system. {x+y+z=4x−2y+z=7x+3y+2z=4\(\begin{cases}\)x + y + z = 4 \(\x\) - 2y + z = 7 \(\x\) + 3y + 2z = 4\(\end{cases}\)

Verified step by step guidance
1
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, \(A = \begin{bmatrix} 1 & 1 & 1 \\ 1 & -2 & 1 \\ 1 & 3 & 2 \end{bmatrix}\), \(\mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix}\), and \(\mathbf{b} = \begin{bmatrix} 4 \\ 7 \\ 4 \end{bmatrix}\).
Calculate the determinant of the coefficient matrix \(A\), denoted as \(\det(A)\). This is essential because Cramer's Rule requires \(\det(A) \neq 0\) to have a unique solution.
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 each variable: \(x = \frac{\det(A_x)}{\det(A)}\), \(y = \frac{\det(A_y)}{\det(A)}\), and \(z = \frac{\det(A_z)}{\det(A)}\). These fractions give the solution to the system.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
8m
Was this helpful?

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 coefficient matrix has a non-zero determinant. Each variable is found by replacing the corresponding column in the coefficient matrix with the constants vector and calculating the determinant ratio.
Recommended video:
Guided course
6:54
Cramer's Rule - 2 Equations with 2 Unknowns

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.
Recommended video:
Guided course
4:36
Determinants of 2×2 Matrices

Systems of Linear Equations

A system of linear equations consists of multiple linear equations with the same 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 useful for small systems.
Recommended video:
Guided course
4:27
Introduction to Systems of Linear Equations
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\(\end{bmatrix}\)

A - C

Textbook Question

Solve the system: (Hint: Let A = ln w, B = ln x, C = ln y, and D = ln z. Solve the system for A, B, C, and D. Then use the logarithmic equations to find w, x, y, and z.)

{2lnw+lnx+3lny−2lnz=−64lnw+3lnx+lny−lnz=−2lnw+lnx+lny+lnz=−5lnw+lnx−lny−lnz=5\(\begin{cases}\)2 \(\ln\) w + \(\ln\) x + 3 \(\ln\) y - 2 \(\ln\) z = -6 \\4 \(\ln\) w + 3 \(\ln\) x + \(\ln\) y - \(\ln\) z = -2 \(\ln\) w + \(\ln\) x + \(\ln\) y + \(\ln\) z = -5 \(\ln\) w + \(\ln\) x - \(\ln\) y - \(\ln\) z = 5\(\end{cases}\)

4
views
Textbook Question

In Exercises 37–44, use Cramer's Rule to solve each system. {x+2z=42y−z=52x+3y=13\(\begin{cases}\)x + 2z = 4 \\2y - z = 5 \\2x + 3y = 13\(\end{cases}\)

5
views
Textbook Question

In Exercises 39–42, find A^(-1) Check that AA^-1 = I and A^(-1)A = I

Textbook Question

Find the cubic function f(x) = ax³ + bx² + cx + d for which ƒ( − 1) = 0, ƒ(1) = 2, ƒ(2) = 3, and ƒ(3) = 12.

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.

{w−x+2y=−3x−y+z=4−w+x−y+2z=2−x+y−2z=−4The inverse of [1−12001−11−11−120−11−2] is [00−1−1141312120−10−1]\(\begin{cases}\)w - x + 2y \(\quad\]\quad\) = -3 \(\quad\[\quad\) x - y + z = 4 \\-w + x - y + 2z = 2 \(\quad\]\quad\) -x + y - 2z = -4\(\end{cases}\[\text{The inverse of }\]\begin{bmatrix}\)1 & -1 & 2 & 0 \\0 & 1 & -1 & 1 \\-1 & 1 & -1 & 2 \\0 & -1 & 1 & -2\(\end{bmatrix}\[\text{ is }\]\begin{bmatrix}\)0 & 0 & -1 & -1 \\1 & 4 & 1 & 3 \\1 & 2 & 1 & 2 \\0 & -1 & 0 & -1\(\end{bmatrix}\)