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

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Step 1: Identify the matrices A and B given in the problem. Matrix A is $ \begin{bmatrix} 2 \\ -4 \\ 1 \end{bmatrix} $ and matrix B is $ \begin{bmatrix} -5 \\ 3 \\ -1 \end{bmatrix} $.

Step 2: Understand the operation to perform: $ -3A + 2B $. This means you will multiply each element of matrix A by -3 and each element of matrix B by 2.

Step 3: Multiply matrix A by -3. Multiply each element of A by -3: $ -3 \times 2 $, $ -3 \times -4 $, and $ -3 \times 1 $.

Step 4: Multiply matrix B by 2. Multiply each element of B by 2: $ 2 \times -5 $, $ 2 \times 3 $, and $ 2 \times -1 $.

Step 5: Add the resulting matrices from steps 3 and 4 element-wise. That is, add the corresponding elements from the two matrices to get the final matrix $ -3A + 2B $.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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. These 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. Here, both A and B are 3x1 matrices, so operations like -3A + 2B are valid. Understanding dimensions ensures correct application of matrix operations.

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 order ensures accurate computation and avoids errors in combining matrices.

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