Perform each operation, if possible. [−1025][−3428]\left[ \begin{matrix} -1 & 0 \\ 2 & 5 \end{matrix} \right] \left[ \begin{matrix} -3 & 4 \\ 2 & 8 \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 77

6장, 문제 77

Perform each operation, if possible.
$[\begin{matrix} - 1 & 0 \\ 2 & 5 \end{matrix}] [\begin{matrix} - 3 & 4 \\ 2 & 8 \end{matrix}]$

검증된 단계별 안내

Identify the two 2x2 matrices given in the problem. Let's call the first matrix $ A = \begin{bmatrix} a & b \\ c & d \end{bmatrix} $ and the second matrix $ B = \begin{bmatrix} e & f \\ g & h \end{bmatrix} $.

Determine the operation to perform between the two matrices. Common operations include addition, subtraction, and multiplication. Confirm which operation is requested.

If the operation is addition or subtraction, add or subtract the corresponding elements of the matrices: $ A \pm B = \begin{bmatrix} a \pm e & b \pm f \\ c \pm g & d \pm h \end{bmatrix} $.

If the operation is multiplication, multiply the matrices using the rule for matrix multiplication: $ (AB)_{ij} = \sum_{k=1}^2 A_{ik} B_{kj} $. Specifically, calculate each element of the product matrix as follows:

$ \begin{bmatrix} a \times e + b \times g & a \times f + b \times h \\ c \times e + d \times g & c \times f + d \times h \end{bmatrix} $.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

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

Matrix Dimensions and Compatibility

Understanding the size of matrices (rows × columns) is essential to determine if operations like addition, subtraction, or multiplication are possible. For addition and subtraction, matrices must have identical dimensions. For multiplication, the number of columns in the first matrix must equal the number of rows in the second.

Matrix Addition and Subtraction

Matrix addition and subtraction involve combining corresponding elements from two matrices of the same size. Each element in the resulting matrix is the sum or difference of elements in the same position from the original matrices.

Matrix Multiplication

Matrix multiplication involves taking the dot product of rows from the first matrix with columns from the second. The resulting matrix has dimensions equal to the number of rows of the first matrix and columns of the second. This operation is not element-wise and requires compatible dimensions.

Perform each operation, if possible. [−1025][−3428]\(\left\)[ \(\begin{matrix}\) -1 & 0 \\ 2 & 5 \(\end{matrix}\) \(\right\)] \(\left\)[ \(\begin{matrix}\) -3 & 4 \\ 2 & 8 \(\end{matrix}\) \(\right\)]