In Exercises 9 - 16, find the following matrices: d. - 3A + 2B 3 1 1 2 - 3 6 A = B = - 1 2 5 - 3 1 - 4

Verified step by step guidance

1. Identify the matrices A and B from the problem statement. Matrix A is \( \begin{bmatrix} 3 & 1 & 1 \\ 2 & -1 & 2 \end{bmatrix} \) and matrix B is \( \begin{bmatrix} -3 & 6 \\ 1 & -4 \end{bmatrix} \).
2. Multiply matrix A by -3. This involves multiplying each element of matrix A by -3.
3. Multiply matrix B by 2. This involves multiplying each element of matrix B by 2.
4. Add the resulting matrix from step 2 to the resulting matrix from step 3. Ensure that the matrices are of the same dimensions before adding.
5. The resulting matrix from step 4 is the solution to the expression \(-3A + 2B\).

Key Concepts

Matrix Addition and Scalar Multiplication

Matrix addition involves combining two matrices of the same dimensions by adding their corresponding elements. Scalar multiplication refers to multiplying each element of a matrix by a constant (scalar). In the expression -3A + 2B, we first multiply matrix A by -3 and matrix B by 2, and then we add the resulting matrices together.

Matrix Representation

Matrices are rectangular arrays of numbers arranged in rows and columns. Each element in a matrix is identified by its position, typically denoted as A[i][j], where i is the row index and j is the column index. Understanding how to represent and manipulate matrices is crucial for performing operations like addition and scalar multiplication.

Order of Operations in Matrix Algebra

In matrix algebra, the order of operations is important, similar to arithmetic. When performing operations like -3A + 2B, we must first apply the scalar multiplications before performing the addition. This ensures that the calculations are done correctly and that the resulting matrix is accurate.