In Exercises 9 - 16, find the following matrices: d. - 3A + 2B

검증된 단계별 안내

Step 1: Identify the matrices A and B from the problem. Matrix A is \( \begin{bmatrix} 1 & 3 \\ 3 & 4 \\ 5 & 6 \end{bmatrix} \) and matrix B is \( \begin{bmatrix} 2 & -1 \\ 3 & -2 \\ 0 & 1 \end{bmatrix} \).

Step 2: Multiply matrix A by -3. This means multiplying each element of matrix A by -3, resulting in a new matrix \( -3A \).

Step 3: Multiply matrix B by 2. This means multiplying each element of matrix B by 2, resulting in a new matrix \( 2B \).

Step 4: Add the two resulting matrices from Step 2 and Step 3 element-wise. That is, add corresponding elements from \( -3A \) and \( 2B \) to get the matrix \( -3A + 2B \).

Step 5: Verify that the resulting matrix has the same dimensions as the original matrices (3 rows and 2 columns) and write the final matrix as the answer.

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 Addition and Scalar Multiplication

Matrix addition involves adding corresponding elements of two matrices of the same dimensions. Scalar multiplication means multiplying every element of a matrix by a constant. Both operations are fundamental for combining matrices as in the expression -3A + 2B.

Matrix Dimensions and Compatibility

For matrix addition or subtraction, the matrices must have the same dimensions (same number of rows and columns). Here, matrices A and B are both 3x2, so operations like -3A + 2B are valid and can be performed element-wise.

Order of Operations in Matrix Expressions

When evaluating expressions like -3A + 2B, scalar multiplication is performed first on each matrix, followed by matrix addition. This ensures correct calculation by applying multiplication before addition, similar to arithmetic order of operations.