Question

If A^T = `[(4, 5),(-1, 0),(2, 3)]` and B = `[(2, -1, 1),(7, 5, -2)]`, veriy the following 

(A + B)^T = A^T + B^T = B^T + A^T

Answer 1

A^T = `[(4, 5),(-1, 0),(2, 3)]` and B = `[(2, -1, 1),(7, 5, -2)]`

A^T = `[(4, 5),(-1, 0),(2, 3)]`

(A^T)^T = `[(4, 5),(-1, 0),(2, 3)]^"T"`

A = `[(4, -1, 2),(5, 0, 3)]`

B^T = `[(2, 7),(-1, 5),(1, -2)]`

A + B = `[(4, -1, 2),(5, 0, 3)]+ [(2, -1, 1),(7, 5, -2)]`

A + B = `[(4 + 2, -1 - 1, 2 +1),(5 + 7,  + 5, 3 -2)]`

= `[(6, -2, 3),(12, 5, 1)]`

(A + B)^T = `[(6, 12),(-2, 5),(3, 1)]`  .......(1)

A^T + B^T = `[(4, 5),(-1, 0),(2, 3)] [(2, 7),(-1, 5),(1, -2)]`

= `[(4 + 2, 5 +7),(-1 - 1, 0 + 5),(2 + 1, 3 - 2)]`

(A + B)^T = `[(6, 12),(-2, 5),(3, 1)]`  .......(2)

B^T + A^T = `[(2, 7),(-1, 5),(1, -2)] + [(4, 5),(-1, 0),(2, 3)]`

= `[(2 + 4, 7 + 5),(-1 - 1, 5 + 0),(1 + 2, -2 + 3)]`

B^T + A^T = `[(6, 12),(-2, 5),(3, 1)]`  .......(3)

From eqution (1), (2) and (3)

(A + B)^T = A^T + B^T = B^T + A^T