A^T + B^T = (A + B)^T
True. See properties of transposition. Also should be able to think through to show this. When we add corresponding entries, these will remain corresponding entries after transposition. This is mentioned in Theorem 3B.
The statement A^T + B^T = (A + B)^T is true for any matrices A and B of compatible sizes.
To see why this is true, consider the transpose of the sum of two matrices A and B:
(A + B)^T
By the definition of matrix addition and transpose, we have:
(A + B)^T = (a_ij + b_ij)^T = (a_ij)^T + (b_ij)^T = A^T + B^T
where the last equality follows from the definition of matrix transpose, which states that the transpose of a sum of matrices is the sum of their transposes.
Therefore, we have shown that A^T + B^T = (A + B)^T for any matrices A and B of compatible sizes, and thus the statement is true.
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