The transpose of a sum of matrices equals the sum of their transposes.
True according to Theorem 3B which says that the transpose of a sum of matrices equals the sum of their transposes.
The transpose of a matrix is found by interchanging its rows and columns. So, if we have two matrices A and B, and C is their sum, then their transpose can be defined as follows:
Transpose of A: (AT)ij = Aji
Transpose of B: (BT)ij = Bji
Now, let’s consider the transpose of their sum:
(CT)ij = Cji
Where Cij = Aij + Bij
We can now see that:
(CT)ij = Cji = (Aij + Bij)T = Aji + Bji = (AT)ij + (BT)ij
Therefore, we just proved that the transpose of a sum of matrices equals the sum of their transposes.
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