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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 35ab
7장, 문제 35ab

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 2 2 - 3 1 - 1 - 1 1 A = B = 1 1 - 2 1 5 4 10 5
Matrices A and B for matrix operations in college algebra, chapter on matrices and determinants.

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Step 1: Identify the dimensions of matrices A and B. Matrix A is a 2x4 matrix (2 rows, 4 columns), and matrix B is a 4x2 matrix (4 rows, 2 columns).
Step 2: To find the product AB, check if the number of columns in A equals the number of rows in B. Since A is 2x4 and B is 4x2, multiplication AB is possible and the resulting matrix will be 2x2.
Step 3: To find the product BA, check if the number of columns in B equals the number of rows in A. Since B is 4x2 and A is 2x4, multiplication BA is possible and the resulting matrix will be 4x4.
Step 4: Calculate each element of the product AB by taking the dot product of the rows of A with the columns of B. For example, the element in the first row and first column of AB is calculated as: A[1,1] * B[1,1] + A[1,2] * B[2,1] + A[1,3] * B[3,1] + A[1,4] * B[4,1].
Step 5: Similarly, calculate each element of the product BA by taking the dot product of the rows of B with the columns of A. For example, the element in the first row and first column of BA is calculated as: B[1,1] * A[1,1] + B[1,2] * A[2,1]. Repeat this for all elements to fill the 4x4 matrix.

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주요 개념

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

Matrix Multiplication

Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. The number of columns in the first matrix must equal the number of rows in the second matrix for the product to be defined.
추천 영상:
03:42
Finding Zeros & Their Multiplicity

Dimension Compatibility

For two matrices A and B, the product AB is defined only if the number of columns in A equals the number of rows in B. Similarly, BA is defined only if the number of columns in B equals the number of rows in A.

Non-Commutativity of Matrix Multiplication

Matrix multiplication is generally not commutative, meaning AB does not necessarily equal BA. Both products may exist or only one may be defined, and their results can be different matrices.
추천 영상:
6:37
Permutations of Non-Distinct Objects
관련 실천
교과서 질문

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\(\end{bmatrix}\)

4B - 3C

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교과서 질문

Write each matrix equation as a system of linear equations without matrices.

[20−1030110][xyz]=[695]\(\begin{bmatrix}\) 2 & 0 & -1 \\ 0 & 3 & 0 \\ 1 & 1 & 0 \(\end{bmatrix}\) \(\begin{bmatrix}\) x \\ y \\ z \(\end{bmatrix}\) = \(\begin{bmatrix}\) 6 \\ 9 \\ 5 \(\end{bmatrix}\)

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교과서 질문

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. ∣0.5750.5390.513∣\(\begin{vmatrix}\) 0.5 & 7 & 5 \\ 0.5 & 3 & 9 \\ 0.5 & 1 & 3 \(\end{vmatrix}\)

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교과서 질문

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

{w+x+y+z=42w+x−2y−z=0w−2x−y−2z=−23w+2x+y+3z=4\(\begin{cases}\) w + x + y + z = 4 \\ 2w + x - 2y - z = 0 \\ w - 2x - y - 2z = -2 \\ 3w + 2x + y + 3z = 4 \(\end{cases}\)

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교과서 질문

In Exercises 37–44, use Cramer's Rule to solve each system. {x+y+z=02x−y+z=−1−x+3y−z=−8\(\begin{cases}\) x + y + z = 0 \\ 2x - y + z = -1 \\ -x + 3y - z = -8 \(\end{cases}\)

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교과서 질문

In Exercises 37–38, find the products and to determine whether B is the multiplicative inverse of A.

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