Skip to main content
Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblema 17
Capitolo 7, Problema 17

For Exercises 11–22, use Cramer's Rule to solve each system. {x+2y=33x−4y=4\(\begin{cases}\) x + 2y = 3 \\ 3x - 4y = 4 \(\end{cases}\)

Guida verificata passo dopo passo
1
Write the system of equations in matrix form: \( A\mathbf{x} = \mathbf{b} \), where \( A = \begin{bmatrix} 1 & 2 \\ 3 & -4 \end{bmatrix} \), \( \mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix} \), and \( \mathbf{b} = \begin{bmatrix} 3 \\ 4 \end{bmatrix} \).
Calculate the determinant of matrix \( A \), denoted as \( \det(A) \), using the formula for a 2x2 matrix: \( \det(A) = a_{11}a_{22} - a_{12}a_{21} \). For this matrix, \( \det(A) = (1)(-4) - (2)(3) \).
Form matrix \( A_x \) by replacing the first column of \( A \) with vector \( \mathbf{b} \), so \( A_x = \begin{bmatrix} 3 & 2 \\ 4 & -4 \end{bmatrix} \). Then calculate \( \det(A_x) \) using the same determinant formula.
Form matrix \( A_y \) by replacing the second column of \( A \) with vector \( \mathbf{b} \), so \( A_y = \begin{bmatrix} 1 & 3 \\ 3 & 4 \end{bmatrix} \). Then calculate \( \det(A_y) \) using the determinant formula.
Use Cramer's Rule to find the solutions: \( x = \frac{\det(A_x)}{\det(A)} \) and \( y = \frac{\det(A_y)}{\det(A)} \).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
4m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.
Video consigliato:
6:54
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.
Video consigliato:
4:36
Determinants of 2×2 Matrices

Solving Systems of Linear Equations

Solving systems of linear equations involves finding values for variables that satisfy all equations simultaneously. Methods include substitution, elimination, and matrix approaches like Cramer's Rule. Understanding how to manipulate and interpret equations is essential for applying these methods effectively.
Video consigliato:
4:27
Introduction to Systems of Linear Equations
Pratica correlata
Domanda del libro di testo

In Exercises 9 - 16, find the following matrices: c. - 4A

89
views
Domanda del libro di testo

Use the fact that if A=[abcd]A=\(\begin{bmatrix}\)a & b\\ c & d\(\end{bmatrix}\), then A−1=1ad−bc[d−b−ca]A^{-1}=\(\frac{1}{ad-bc}\)\(\begin{bmatrix}\)d & -b\\ -c & a\(\end{bmatrix}\) to find the inverse of each matrix, if possible. Check that AA−1=I2AA^{-1} = I_2 and A−1A=I2A^{-1}A = I_2.

A=[10−2−51]A = \(\begin{bmatrix}\) 10 & -2 \\ -5 & 1 \(\end{bmatrix}\)

622
views
Domanda del libro di testo

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. D-A

871
views
Domanda del libro di testo

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. X - A = B

74
views
Domanda del libro di testo

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 \(\end{array}\)

937
views
Domanda del libro di testo

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

792
views