Question

For the matrices below, indicate whether the operations listed under (a)-(c) are well defined. If the operation is well defined, compute its result. If not, explain why. A=[04​1−1​−12​],B=[13​2−1​],C=⎣⎡​231​0−1−2​⎦⎤​ (a) AC−B (b) AᵀB−2C (c) BᵀA−C

Answer 1

To ascertain if matrix operations are well defined, dimensions of the matrices must be compatible for multiplication and subtraction. It depends on the matching of rows to columns in matrix multiplication and size equivalence for subtraction.

Step-by-step explanation:

The student has asked whether the matrix operations (a) AC - B, (b) AᵀB - 2C, and (c) BᵀA - C are well defined and to compute the results if they are. To determine if a matrix operation is well defined, we need to consider the dimensions of the matrices involved and whether the operations (such as multiplication or subtraction) are feasible with those dimensions.

1. Matrix Multiplication and Subtraction: For AC - B to be well defined, the number of columns in A must equal the number of rows in C, and the resultant matrix from AC must be the same size as B for subtraction to occur.
2. Matrix Transpose Multiplication and Scalar Multiplication: For AᵀB - 2C to be well defined, the number of rows in A (after taking the transpose, Aᵀ) must equal the number of rows in B, and 2C must be scalar multiplication, where C is multiplied by the scalar 2.
3. Transpose and Subtraction: For BᵀA - C to be well defined, the operation BᵀA requires the number of columns in B after taking the transpose (Bᵀ) to equal the number of rows in A, and the resultant matrix must be the same size as C for subtraction to occur.