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

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Step 1: Identify the matrices A and B. Matrix A is $ \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix} $ and matrix B is $ \begin{bmatrix} 5 & 9 \\ 0 & 7 \end{bmatrix} $.

Step 2: Multiply matrix A by -3. This means multiplying each element of matrix A by -3, resulting in $ -3A = \begin{bmatrix} -3 \times 4 & -3 \times 1 \\ -3 \times 3 & -3 \times 2 \end{bmatrix} $.

Step 3: Multiply matrix B by 2. This means multiplying each element of matrix B by 2, resulting in $ 2B = \begin{bmatrix} 2 \times 5 & 2 \times 9 \\ 2 \times 0 & 2 \times 7 \end{bmatrix} $.

Step 4: Add the resulting matrices from Step 2 and Step 3 element-wise. That is, add corresponding elements from $ -3A $ and $ 2B $ to get $ -3A + 2B = \begin{bmatrix} (-3 \times 4) + (2 \times 5) & (-3 \times 1) + (2 \times 9) \\ (-3 \times 3) + (2 \times 0) & (-3 \times 2) + (2 \times 7) \end{bmatrix} $.

Step 5: Write the resulting matrix from Step 4 as the final answer for $ -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 from two matrices of the same dimensions. Scalar multiplication means multiplying every element of a matrix by a constant. In the expression -3A + 2B, you first multiply each element of A by -3 and each element of B by 2, then add the resulting matrices element-wise.

Matrix Dimensions and Compatibility

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

Order of Operations in Matrix Expressions

When evaluating expressions like -3A + 2B, follow the order of operations: first perform scalar multiplications (-3A and 2B), then add the resulting matrices. This systematic approach prevents errors and ensures accurate computation of the final matrix.

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