In Exercises 9 - 16, find the following matrices: a. A + B

Verified step by step guidance

1. Step 1: Identify the matrices A and B. Matrix A is \( \begin{bmatrix} 3 & 1 & 1 \\ -1 & 2 & 5 \end{bmatrix} \) and matrix B is \( \begin{bmatrix} 2 & -3 & 6 \\ -3 & 1 & -4 \end{bmatrix} \).
2. Step 2: Confirm that both matrices have the same dimensions. Here, both A and B are 2x3 matrices, so addition is possible.
3. Step 3: Add the corresponding elements of matrices A and B. This means adding each element in the first row of A to the corresponding element in the first row of B, and similarly for the second row.
4. Step 4: Write the resulting matrix from the addition. For example, the element in the first row and first column of the result is \(3 + 2\), the element in the first row and second column is \(1 + (-3)\), and so on.
5. Step 5: Express the final matrix as \( A + B = \begin{bmatrix} 3+2 & 1+(-3) & 1+6 \\ -1+(-3) & 2+1 & 5+(-4) \end{bmatrix} \).

Key Concepts

Matrix Addition

Matrix addition involves adding corresponding elements from two matrices of the same dimensions. Each element in the resulting matrix is the sum of elements in the same position from the original matrices. This operation is only defined when both matrices have the same number of rows and columns.

Matrix Dimensions

The dimensions of a matrix are given by the number of rows and columns it contains. For two matrices to be added, they must have identical dimensions. In this problem, both matrices A and B are 2x3 matrices, making addition possible.

Element-wise Operations

Element-wise operations apply a specific operation, such as addition or subtraction, to each corresponding element of matrices. Understanding this concept is crucial for correctly performing matrix addition, ensuring each element is combined with its counterpart.