Given A=[4−231],B=[510−237]A = \left[ \begin{matrix} 4 & -2 \\ 3 & 1 \end{matrix} \right], \quad B = \left[ \begin{matrix} 5 & 1 \\ 0 & -2 \\ 3 & 7 \end{matrix} \right] , and C=[−541036]C = \left[ \begin{matrix} -5 & 4 & 1 \\ 0 & 3 & 6 \end{matrix} \right] C=[−504316], find each product, if possible. See Examples 5–7. AB

Ch. 5 - Systems and Matrices

Lial13th EditionCollege AlgebraISBN: 9780136881063당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 73

6장, 문제 73

Given $A = [\begin{matrix} 4 & - 2 \\ 3 & 1 \end{matrix}],$ , and $C = $\left$[ $\begin{matrix}$ -5 & 4 & 1 \\ 0 & 3 & 6 $\end{matrix}$ $\right$]$ , find each product, if possible. See Examples 5–7. AB

검증된 단계별 안내

Identify the dimensions of matrices A and B. For matrix multiplication AB to be defined, the number of columns in A must equal the number of rows in B.

Write down the dimensions of A and B explicitly. For example, if A is an m×n matrix and B is a p×q matrix, then n must equal p for AB to be possible.

If the multiplication is possible, set up the product matrix AB, which will have dimensions m×q.

Calculate each element of the product matrix AB by taking the dot product of the corresponding row of A with the corresponding column of B. Specifically, the element in row i and column j of AB is given by $\sum_{k=1}^n A_{ik} \times B_{kj}$.

Perform the multiplication for each element systematically to fill the entire product matrix AB.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Matrix Dimensions and Compatibility

Matrix multiplication is only defined when the number of columns in the first matrix equals the number of rows in the second matrix. For example, if A is an m×n matrix and B is a p×q matrix, the product AB exists only if n = p.

Matrix Multiplication Procedure

To multiply two matrices, each element of the resulting matrix is computed as the dot product of the corresponding row from the first matrix and the column from the second matrix. This involves multiplying corresponding entries and summing the results.

Resulting Matrix Dimensions

The product of an m×n matrix and an n×p matrix results in an m×p matrix. Understanding the size of the resulting matrix helps in verifying the correctness of the multiplication and organizing the computation.