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

Verified step by step guidance

1. Step 1: Identify the matrices A and B. Matrix A is \( \begin{bmatrix} 4 & 1 \\ 3 & 2 \end{bmatrix} \) and matrix B is \( \begin{bmatrix} 5 & 9 \\ 0 & 7 \end{bmatrix} \).
2. Step 2: Multiply matrix A by -3. This means multiplying each element of matrix A by -3, resulting in \( -3A = \begin{bmatrix} -3 \times 4 & -3 \times 1 \\ -3 \times 3 & -3 \times 2 \end{bmatrix} \).
3. Step 3: Multiply matrix B by 2. This means multiplying each element of matrix B by 2, resulting in \( 2B = \begin{bmatrix} 2 \times 5 & 2 \times 9 \\ 2 \times 0 & 2 \times 7 \end{bmatrix} \).
4. 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 \( -3A + 2B = \begin{bmatrix} (-3 \times 4) + (2 \times 5) & (-3 \times 1) + (2 \times 9) \\ (-3 \times 3) + (2 \times 0) & (-3 \times 2) + (2 \times 7) \end{bmatrix} \).
5. Step 5: Write the resulting matrix from Step 4 as the final answer for \( -3A + 2B \).

Key Concepts

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.