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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Blitzer 8th EditionCh. 6 - Matrices and Determinants문제 15
7장, 문제 15

For Exercises 11–22, use Cramer's Rule to solve each system. {4x−5y=172x+3y=3\(\begin{cases}\) 4x - 5y = 17 \\ 2x + 3y = 3 \(\end{cases}\)

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Write the system of equations in matrix form as \(A\mathbf{x} = \mathbf{b}\), where \(A = \begin{bmatrix} 4 & -5 \\ 2 & 3 \end{bmatrix}\), \(\mathbf{x} = \begin{bmatrix} x \\ y \end{bmatrix}\), and \(\mathbf{b} = \begin{bmatrix} 17 \\ 3 \end{bmatrix}\).
Calculate the determinant of matrix \(A\), denoted as \(\det(A)\), using the formula \(\det(A) = a_{11}a_{22} - a_{12}a_{21}\). For this matrix, compute \(\det(A) = 4 \times 3 - (-5) \times 2\).
Form matrix \(A_x\) by replacing the first column of \(A\) with vector \(\mathbf{b}\), so \(A_x = \begin{bmatrix} 17 & -5 \\ 3 & 3 \end{bmatrix}\). Then calculate \(\det(A_x)\) using the determinant formula.
Form matrix \(A_y\) by replacing the second column of \(A\) with vector \(\mathbf{b}\), so \(A_y = \begin{bmatrix} 4 & 17 \\ 2 & 3 \end{bmatrix}\). 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 \(y = \frac{\det(A_y)}{\det(A)}\). These fractions give the values of \(x\) and \(y\) that solve the system.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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 determinant of the coefficient matrix is non-zero. Each variable is found by replacing the corresponding column in the coefficient matrix with the constants vector and calculating the determinant ratio.
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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 value helps determine if the system has a unique solution (non-zero determinant) or not. Determinants are essential in Cramer's Rule to find the values of variables by comparing determinants of modified matrices.
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Determinants of 2×2 Matrices

Solving Systems of Linear Equations

A system of linear equations consists of multiple linear equations with the same variables. The goal is to find values for the variables that satisfy all equations simultaneously. Methods include substitution, elimination, and matrix approaches like Cramer's Rule, which is efficient for small systems.
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Introduction to Systems of Linear Equations
관련 실천
교과서 질문

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

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교과서 질문

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=[3−1−42]A = \(\begin{bmatrix}\) 3 & -1 \\ -4 & 2 \(\end{bmatrix}\)

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교과서 질문

Perform each matrix row operation and write the new matrix.

[1−32∣031−1∣72−21∣3]−3R1+R2\(\begin{bmatrix}\) 1 & -3 & 2 & \(\vert\) & 0 \\ 3 & 1 & -1 & \(\vert\) & 7 \\ 2 & -2 & 1 & \(\vert\) & 3 \(\end{bmatrix}\) \(\quad\) -3R_1 + R_2

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교과서 질문

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

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교과서 질문

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

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교과서 질문

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

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