Perform each operation, if possible. [−2550143−4−1][10−1−10011−1]\left[ \begin{matrix} -2 & 5 & 5 \\ 0 & 1 & 4 \\ 3 & -4 & -1 \end{matrix} \right] \left[ \begin{matrix} 1 & 0 & -1 \\ -1 & 0 & 0 \\ 1 & 1 & -1 \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 81

6장, 문제 81

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

검증된 단계별 안내

Identify the operation to be performed between the two 3x3 matrices. Common operations include addition, subtraction, and multiplication. Confirm which operation is requested.

If the operation is addition or subtraction, verify that both matrices have the same dimensions (which they do, both are 3x3). Then, add or subtract corresponding elements from each matrix. For example, if the first matrix element is $a_{ij}$ and the second is $b_{ij}$, the result element is $c_{ij} = a_{ij} \pm b_{ij}$.

If the operation is multiplication, recall that the product of two 3x3 matrices $A$ and $B$ is another 3x3 matrix $C$, where each element $c_{ij}$ is calculated by taking the dot product of the $i$-th row of $A$ with the $j$-th column of $B$. Mathematically, $c_{ij} = \sum_{k=1}^3 a_{ik} b_{kj}$.

To compute each element $c_{ij}$ in the product matrix, multiply corresponding elements from the $i$-th row of the first matrix and the $j$-th column of the second matrix, then sum these products. Repeat this for all $i$ and $j$ from 1 to 3.

After computing all elements, assemble them into the resulting 3x3 matrix. This matrix is the final result of the operation.

비슷한 문제에 대한 검증된 영상 답변:

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주요 개념

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

Matrix Addition and Subtraction

Matrix addition and subtraction involve combining two matrices of the same dimensions by adding or subtracting their corresponding elements. Each element in the resulting matrix is found by performing the operation on elements in the same position from both matrices.

Matrix Multiplication

Matrix multiplication requires that the number of columns in the first matrix equals the number of rows in the second. The product matrix's elements are calculated by taking the dot product of rows from the first matrix with columns from the second, summing the products of corresponding entries.

Dimension Compatibility

Operations on matrices are only possible when their dimensions meet specific criteria: addition and subtraction require identical dimensions, while multiplication requires the first matrix's columns to match the second matrix's rows. Understanding these rules ensures the operation is valid.

Perform each operation, if possible. [−2550143−4−1][10−1−10011−1]\(\left\)[ \(\begin{matrix}\) -2 & 5 & 5 \\ 0 & 1 & 4 \\ 3 & -4 & -1 \(\end{matrix}\) \(\right\)] \(\left\)[ \(\begin{matrix}\) 1 & 0 & -1 \\ -1 & 0 & 0 \\ 1 & 1 & -1 \(\end{matrix}\) \(\right\)]