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)

검증된 단계별 안내

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.

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]].

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]].

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.

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.

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주요 개념

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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.