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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 21
7장, 문제 21

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. 3X + 2A = B

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Start with the given matrix equation: \(3X + 2A = B\).
Isolate the term with \(X\) by subtracting \$2A$ from both sides: \(3X = B - 2A\).
To solve for \(X\), divide both sides of the equation by 3, which is equivalent to multiplying by \(\frac{1}{3}\): \(X = \frac{1}{3}(B - 2A)\).
Calculate the matrix \$2A\( by multiplying each element of matrix \)A$ by 2.
Subtract the matrix \$2A\( from matrix \)B\( element-wise, then multiply the resulting matrix by \(\frac{1}{3}\) to find matrix \)X$.

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주요 개념

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

Matrix Addition and Scalar Multiplication

Matrix addition involves adding corresponding elements of two matrices of the same size. Scalar multiplication means multiplying every element of a matrix by a constant. These operations are essential to manipulate the equation 3X + 2A = B by distributing scalars and combining matrices.
추천 영상:
03:42
Finding Zeros & Their Multiplicity

Solving Matrix Equations

To solve matrix equations like 3X + 2A = B, isolate the matrix variable X by performing inverse operations. This typically involves subtracting 2A from both sides and then multiplying by the inverse of the scalar coefficient (here, dividing by 3) to find X.
추천 영상:
5:02
Solving Logarithmic Equations

Matrix Dimensions and Compatibility

Matrix operations require matrices to have compatible dimensions. Both A and B are 3x2 matrices, so X must also be 3x2 for the equation to be valid. Understanding dimensions ensures correct addition and scalar multiplication without errors.
추천 영상:
4:35
Introduction to Matrices
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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+x−y+z=−22w−x+2y−z=7−w+2x+y+2z=−1\(\begin{cases}\) w + x - y + z = -2 \\ 2w - x + 2y - z = 7 \\ -w + 2x + y + 2z = -1 \(\end{cases}\)

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교과서 질문

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. -5(A+D)

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교과서 질문

For Exercises 11–22, use Cramer's Rule to solve each system. {2x=3y+25x=51−4y\(\begin{cases}\) 2x = 3y + 2 \\ 5x = 51 - 4y \(\end{cases}\)

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교과서 질문

In Exercises 23–30, use expansion by minors to evaluate each determinant. ∣30021−525−1∣\(\begin{vmatrix}\) 3 & 0 & 0 \\ 2 & 1 & -5 \\ 2 & 5 & -1 \(\end{vmatrix}\)

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교과서 질문

Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

{x+y−z=−22x−y+z=5−x+2y+2z=1\(\begin{cases}\) x + y - z = -2 \\ 2x - y + z = 5 \\ -x + 2y + 2z = 1 \(\end{cases}\)

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교과서 질문

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. 2X + A = B

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