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. -5(A+D)

Verified step by step guidance

1. Step 1: Understand the problem. The problem involves performing matrix operations. You are tasked with calculating -5(A + D), where A and D are matrices. Ensure that the matrices A and D are defined and have the same dimensions, as matrix addition is only defined for matrices of the same size.
2. Step 2: Add the matrices A and D. To do this, add the corresponding elements of matrices A and D. For example, if A = [[a11, a12], [a21, a22]] and D = [[d11, d12], [d21, d22]], then A + D = [[a11 + d11, a12 + d12], [a21 + d21, a22 + d22]].
3. Step 3: Multiply the resulting matrix (A + D) by the scalar -5. To perform scalar multiplication, multiply each element of the matrix (A + D) by -5. For example, if A + D = [[x11, x12], [x21, x22]], then -5(A + D) = [[-5 * x11, -5 * x12], [-5 * x21, -5 * x22]].
4. Step 4: Verify your calculations. Double-check that the addition of matrices A and D was performed correctly and that each element of the resulting matrix was multiplied by -5 accurately.
5. Step 5: If the dimensions of A and D are not the same, state that the operation is not defined because matrix addition requires matrices to have the same dimensions.

Key Concepts

Matrix Addition

Matrix addition involves combining two matrices of the same dimensions by adding their corresponding elements. For example, if A and D are both 2x2 matrices, the sum A + D is obtained by adding each element in A to the corresponding element in D. This operation is only defined when the matrices have the same size.

Scalar Multiplication

Scalar multiplication refers to the process of multiplying each element of a matrix by a scalar (a single number). In the expression -5(A + D), the result of the matrix addition A + D is multiplied by -5, which scales each element of the resulting matrix by -5. This operation is defined for any matrix regardless of its dimensions.

Matrix Dimensions

Matrix dimensions describe the size of a matrix in terms of rows and columns, denoted as 'm x n' where m is the number of rows and n is the number of columns. Understanding dimensions is crucial for determining whether matrix operations, such as addition or multiplication, can be performed. If A and D have different dimensions, the operation A + D would be undefined.