Find each product, if possible. [3−41502][−142]\left[ \begin{matrix} 3 & -4 & 1 \\ 5 & 0 & 2 \end{matrix} \right] \left[ \begin{matrix} -1 \\ 4 \\ 2 \end{matrix} \right]

Ch. 5 - Systems and Matrices

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

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Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 57

6장, 문제 57

Find each product, if possible.
$[\begin{matrix} 3 & - 4 & 1 \\ 5 & 0 & 2 \end{matrix}] [\begin{matrix} - 1 \\ 4 \\ 2 \end{matrix}]$

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Identify the dimensions of the matrices involved. The first matrix is a 3x2 matrix (3 rows and 2 columns), and the second matrix is a 1x3 matrix (1 row and 3 columns).

Recall the rule for matrix multiplication: The number of columns in the first matrix must equal the number of rows in the second matrix for the product to be defined.

Check if the multiplication is possible by comparing the dimensions: The first matrix has 2 columns, and the second matrix has 1 row. Since 2 ≠ 1, the multiplication is not possible.

Conclude that the product of a 3x2 matrix and a 1x3 matrix is undefined because the inner dimensions do not match.

If you want to multiply matrices, consider switching the order or using matrices with compatible dimensions, such as multiplying a 3x2 matrix by a 2xN matrix.

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주요 개념

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Matrix Dimensions and Multiplication Rules

Matrix multiplication is defined only when the number of columns in the first matrix equals the number of rows in the second matrix. For example, a 3x2 matrix can only be multiplied by a 2xN matrix. Understanding these dimension requirements is essential to determine if the product is possible.

Matrix Multiplication Process

To multiply matrices, each element of the resulting matrix is found by taking 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, producing a new matrix with dimensions based on the outer dimensions of the factors.

Interpreting Matrix Size Notation

Matrix size is denoted as rows x columns (e.g., 3x2 means 3 rows and 2 columns). Recognizing this notation helps in visualizing the matrix structure and applying multiplication rules correctly, ensuring the operation is valid and the resulting matrix dimensions are understood.

Find each product, if possible. [3−41502][−142]\(\left\)[ \(\begin{matrix}\) 3 & -4 & 1 \\ 5 & 0 & 2 \(\end{matrix}\) \(\right\)] \(\left\)[ \(\begin{matrix}\) -1 \\ 4 \\ 2 \(\end{matrix}\) \(\right\)]