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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 13
7장, 문제 13

Perform each matrix row operation and write the new matrix.
[2−64∣1015−5∣0304∣7]12R1\(\begin{bmatrix}\)2 & -6 & 4 & \(\vert\) & 10 \\1 & 5 & -5 & \(\vert\) & 0 \\3 & 0 & 4 & \(\vert\) & 7\(\end{bmatrix}\]\quad\) \(\frac{1}{2}\)R_1

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Identify the given matrix and the row operation to be performed. The matrix is: \[\left[ \begin{array}{ccc|c} 2 & -6 & 4 & 10 \\ 1 & 5 & -5 & 0 \\ 3 & 0 & 4 & 7 \end{array} \right]\] and the operation is \[\frac{1}{2} R_1\], which means multiply every element in row 1 by \[\frac{1}{2}\].
Apply the operation to row 1 by multiplying each element in the first row by \[\frac{1}{2}\]: - First element: \[2 \times \frac{1}{2} = 1\] - Second element: \[-6 \times \frac{1}{2} = -3\] - Third element: \[4 \times \frac{1}{2} = 2\] - Augmented element: \[10 \times \frac{1}{2} = 5\]
Write the new matrix with the updated first row and the unchanged rows 2 and 3: \[\left[ \begin{array}{ccc|c} 1 & -3 & 2 & 5 \\ 1 & 5 & -5 & 0 \\ 3 & 0 & 4 & 7 \end{array} \right]\]
Double-check that only row 1 has changed and rows 2 and 3 remain the same.
This completes the row operation \[\frac{1}{2} R_1\] on the matrix.

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주요 개념

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

Matrix Row Operations

Matrix row operations are techniques used to manipulate the rows of a matrix to simplify or solve systems of linear equations. These include row swapping, scaling a row by a nonzero constant, and adding a multiple of one row to another. They preserve the solution set of the system.
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가이드 코스
8:38
Performing Row Operations on Matrices

Scalar Multiplication of a Row

Scalar multiplication involves multiplying every element in a row by the same nonzero constant. This operation changes the row but keeps the system equivalent. For example, multiplying row 1 by 1/2 scales all its entries by 0.5, simplifying the row for further operations.
추천 영상:
가이드 코스
8:38
Performing Row Operations on Matrices

Augmented Matrix Representation

An augmented matrix combines the coefficients of variables and constants from a system of linear equations into one matrix. It helps visualize and perform row operations efficiently to solve the system. The vertical bar separates coefficients from constants.
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가이드 코스
4:35
Introduction to Matrices
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Find the following matrices: A+BA + B

A=[2−41],B=[−53−1]A = \(\begin{bmatrix}\)2 \\-4 \\1\(\end{bmatrix}\), B = \(\begin{bmatrix}\)-5 \\3 \\-1\(\end{bmatrix}\)

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

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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=[23−12]A = \(\begin{bmatrix}\) 2 & 3 \\ -1 & 2 \(\end{bmatrix}\)

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교과서 질문

For Exercises 11–22, use Cramer's Rule to solve each system. {12x+3y=152x−3y=13\(\begin{cases}\)12x + 3y = 15 \\2x - 3y = 13\(\end{cases}\)

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Find the following matrices: - 3A + 2B

A=[311−125],B=[2−36−31−4]A = \(\begin{bmatrix}\)3 & 1 & 1 \\-1 & 2 & 5\(\end{bmatrix}\), B = \(\begin{bmatrix}\)2 & -3 & 6 \\-3 & 1 & -4\(\end{bmatrix}\)

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In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {w−3x+y−4z=4−2w+x+2y=−23w−2x+y−6z=2−w+3x+2y−z=−6\(\begin{cases}\)w - 3x + y - 4z = 4 \\-2w + x + 2y = -2 \\3w - 2x + y - 6z = 2 \\-w + 3x + 2y - z = -6\(\end{cases}\)

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