In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. D-A

Verified step by step guidance

1. Step 1: Understand the problem. The problem involves performing the matrix operation D - A, where D and A are matrices. Matrix subtraction is only defined if the two matrices have the same dimensions. Verify that matrices D and A have the same number of rows and columns.
2. Step 2: Write down the general rule for matrix subtraction. If D and A are matrices of the same dimensions, then the subtraction D - A is performed element by element. Mathematically, this means that for each element in the resulting matrix, (D - A)_{ij} = D_{ij} - A_{ij}, where i and j represent the row and column indices.
3. Step 3: Align the matrices D and A. Write out the elements of both matrices side by side, ensuring that corresponding elements are aligned. This will help you perform the subtraction element by element.
4. Step 4: Subtract the corresponding elements of matrices D and A. For each position (i, j), subtract the element in matrix A from the corresponding element in matrix D. Record the result in the corresponding position of the resulting matrix.
5. Step 5: Verify the result. Ensure that the resulting matrix has the same dimensions as the original matrices D and A, and double-check each subtraction to confirm accuracy. If the dimensions of D and A are not the same, state that the operation is not defined.

Key Concepts

Matrix Operations

Matrix operations include addition, subtraction, and multiplication of matrices. For addition and subtraction, matrices must have the same dimensions, meaning they must have the same number of rows and columns. Understanding these operations is crucial for performing calculations involving matrices, as they follow specific rules that dictate how elements are combined.

Matrix Dimensions

The dimensions of a matrix are defined by the number of rows and columns it contains, expressed as 'm x n' where 'm' is the number of rows and 'n' is the number of columns. When performing operations like addition or subtraction, it is essential that the matrices involved have the same dimensions; otherwise, the operation is undefined. This concept is fundamental in determining whether specific matrix operations can be executed.

Defined Operations

In linear algebra, an operation is considered defined if it adheres to the rules governing the types of matrices involved. For instance, subtracting two matrices is only defined if they have the same dimensions. If the matrices do not meet the criteria for the operation, it is necessary to state that the operation is undefined, which is a critical aspect of matrix algebra.