real numbers:If (a+b)^{2} = a^{2}+b^{2} + 2ab = a^{2}+b^{2}, then 2ab=0 so ab=0 so a=0 or b=0 ( just as you said)

Matrices: If (A+B)^{2} = (A+B)(A+B) = A^{2 }+ AB + BA + B^{2} = A^{2 }+ B^{2}, then AB + BA = 0. Try to figure out the conditions on A and B.

Vectors: Let **V** = < v_{1, }v_{2}, ... ,v_{n}> and **W** = < w_{1, }w_{2}, ... ,w_{n}>. Compute (**V** + **W**)^{2} and find the conditions on **V **and** W **such that it equals **V**^{2} + **W**^{2}

Thank you so much Jomo.

I think that hint for the matrices has really helped get me started:

**Matrices**
Working backwards - the matrices AB and BA can only be added together if they are of the same size (i.e. have the same dimensions).

Now the product matrices AB and BA are always defined when A is mxn AND B is nxm.

BUT the matrices AB and BA will be of different dimensions and not the same size (so contradicting the first line) - i.e. in this case we cannot AB and BA

So A and B must each be matrices of the same dimensions (i.e. m = n)

So A and B are square matrices (in order for AB and BA to be defined).

From there, I believe the only square matrices for A and B that can be satisfied are if either A is the 0 matrix, or if B is the 0 matrix, or if both A and B are the 0 matrix, all with the condition that A and B are of the same dimensions.

Please let me know if this is complete/or have I made a huge error (as I'm usually not great at solving things myself!)

**Vectors**
Again, two vectors of different sizes cannot be added and hence the vectors A and B must be of the same dimensions.

Using the vectors

** a = **< a

_{1, }a

_{2}, ... , a

_{n}> and

**b **= < b

_{1, }b

_{2}, ... ,b

_{n}>

I found that (

**a**+

**b**)

^{2} = < a

_{1}^{2}+2a

_{1}b

_{1}+b

_{2}^{2} , a

_{2}^{2}+2a

_{2}b

_{2}+b

_{2}^{2} ,..., a

_{n}^{2}+2a

_{n}b

_{n}+b

_{n}^{2} >

If (

**a**+

**b**)

^{2} =

**a**^{2} +

**b**^{2}, this vector must equal < a

_{1}^{2}+b

_{2}^{2} , a

_{2}^{2}+b

_{2}^{2} ,..., a

_{n}^{2}+b

_{n}^{2} >

So 2a

_{1}b

_{1}=0, 2a

_{2}b

_{2}=0,.....2a

_{n}b

_{n}=0

So either a

_{k}=0, or b

_{k}=0, or both a

_{k} and b

_{k}=0 (for any integer 1<=k<=n)

Again is this correct?

Finally, is there anything else I could look at other than matrices, vectors and real numbers?

Thank you so much for all your help!!