1.

Let `A=[{:(1,-2),(5," "4),(3," "0):}],B=[{:(" "3," "1),(" "0," "2),(-3," "5):}]" and "C=[{:(" "4," "3),(-2," "2),(" "1," "6):}].` Verify that `(A+B)+C=A+(B+C).`

Answer» Clearly, each one of the martices A, B, C is a `(3xx2)` matrix. So, `(A+B)+C` and `A+(B+C)` are both defined and each one is a `3xx2` matrix.
Now, `(A+B)=[{:(1,-2),(5," "4),(3," "0):}]+[{:(3,1),(0,2),(-3,5):}]`
`=[{:(1+3,-2+1),(5+0," "4+2),(3+(-3)," "0+5):}]+[{:(4,-1),(5," "6),(0," "5):}].`
`:." "(A+B)+C=[{:(4,-1),(5," "6),(0," "5):}]+[{:(" "4," "3),(-2," "2),(" "1," "6):}]`
`=[{:(4+4,-1+3),(5+(-2)," "6+2),(0+1," "5+6):}]=[{:(8,2),(3,8),(1,11):}].`
Also, `(B+C)=[{:(3,1),(0,2),(-3,5):}]+[{:(4,3),(-2,2),(1,6):}]`
`=[{:(3+4,1+3),(0+(-2),2+2),(-3+1,5+6):}]=[{:(7,4),(-2,4),(-2,11):}]`
`:." "A+(B+C)=[{:(1,-2),(5," "4),(3," "0):}]+[{:(7,4),(-2,4),(-2,11):}]`
`=[{:(1+7,-2+4),(5+(-2)," "4+4),(3+(-2)," "0+11):}]=[{:(8,2),(3,8),(1,11):}].`
Hence, `(A+B)+C=A+(B+C).`


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