m times n matrices, A = (aij) and B = (bij), (A+B)^T + B^T?

Question

Please help with the following matrix question - I dont even know where to start. Please show me a step-by-step solution. 
NOTE aij and bij = a (subscript ij) and b (subscript ij).
Thank you

Prove that for an arbitrary pair of m x n (m times n) matrices A= (aij) and B=(bij), (A+B)^T = A^T + B^T. 

Hint: First define C = A + B. Then apply the definition of the transform of a matrix to C and finally unpick that matrix into the separate matrices, again using the definition of the sum of two matrices. You need to write out the various steps and show how the definitions of sum and transpose apply.

efqy · Accepted Answer

The hint tells you step by step how to do it.

? · Answer

If the ij-th term of C is c_ij then the ij-th term of C^T is c_ji

1. the ij-term of A is a_ij, so the ij-th term of A^T is a_ji
    the ij-term of B is b_ij, so the ij-th term of B^T is b_ji

we add matrices term by term, so the ij-th term of A^T+B^T is a_ji+b_ji.

2. the ij-th term of A+B is a_ij+b_ij

so the ij-th term of (A+B)^T is a_ji+b_ji,


the ij-th term of A^T+B^T is the same as the ij-th term of (A+B)^T

so (A+B)^T=A^T+B^T

m times n matrices, A = (aij) and B = (bij), (A+B)^T + B^T?

2 Answers