Question

If A and B are 7 x 4 matrices, and C is a 5 × 7 matrix, which of the following are defined?

A. C+B
B. BᵀCᵀ
C. Cᵀ
D. A-B
E. CB
F. AC

Answer 1

Among the given options, BᵀCᵀ, Cᵀ, and A-B are defined operations; while C+B, CB, and AC are not defined due to incompatible dimensions of the matrices involved.

Step-by-step explanation:

The student question is related to matrix and vector operations, particularly with regards to understanding which operations are defined based on the dimensions of the matrices and vectors involved.

Firstly, we evaluate the possibility of each operation:

1. C + B is not defined because matrices C (5 x 7) and B (7 x 4) do not have the same dimensions.
2. BᵀCᵀ is defined because Bᵀ is a (4 x 7) matrix and Cᵀ is a (7 x 5) matrix, which means they can be multiplied.
3. Cᵀ is always defined as it is the transpose of a (5 x 7) matrix, resulting in a (7 x 5) matrix.
4. A-B is defined because matrices A and B both have the same dimensions (7 x 4).
5. CB is not defined because matrix C is (5 x 7) and B is (7 x 4), which means they can be multiplied to result in a (5 x 4) matrix.
6. AC is not defined because matrix A (7 x 4) cannot be multiplied with matrix C (5 x 7) as the number of columns in A does not match the number of rows in C.