How to prove the transpose property (A+B)^T = A^T + B^T . A and B represent mXn matrices. ^=to the power of.?

Answer 1

Let A =
[a11 … a1n]
[… ]
[am1 … amn]

Let B =
[b11 … b1n]
[… ]
[bm1 … bmn]

Then A+B =
[(a11+b11) … (a1n+b1n)]
[… ]
[(am1+bm1) … (amn+bmn)]

and (A+B)^T =
[(a11+b11) … (am1+bm1)]
[… ]
[(a1n+b1n) … (amn+bmn)]

Let A^T =
[a11 … am1]
[… ]
[a1n … amn]

Let B^T =
[b11 … bm1]
[… ]
[b1n … bmn]

Then (A^T + B^T) =
[(a11+b11) … (am1+bm1)]
[… ]
[(a1n+b1n) … (amn+bmn)]

So (A+B)^T = (A^T + B^T)

**Note that within the matrices, anything that's not an "a" or a "b" should be in subscript (ex. - m, n, 1, 2).**

I hope this makes sense. If you have any questions, don't hesitate to ask.