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## Main Question or Discussion Point

Please help me proove the following:

Let V be a vector space over all n-by-n square matrices. Let W be a non-trivial subspace of V satisfying the following condition: if A is an element of W and B is an element of V then AB, BA are both elements of W.

Proove that W = V.

And here is what I am thinking about it...

1. If W contains the identity matrix then this equivalence is quite obvious. Whatever matrix B from V you give me, I multiply it by the identity matrix I and the B I = B is also element of W.

2. When W contains a regular matrix it is quite similar - because then W must contain the identity matrix. Let A be a regular matrix from W, then if you give me an inverse matrix, when I multiply them then what I will get is the identity matrix and due to the aforesaid condition the identity matrix is an element of W.

But somehow I do not know how to continue.... Please help

Let V be a vector space over all n-by-n square matrices. Let W be a non-trivial subspace of V satisfying the following condition: if A is an element of W and B is an element of V then AB, BA are both elements of W.

Proove that W = V.

And here is what I am thinking about it...

1. If W contains the identity matrix then this equivalence is quite obvious. Whatever matrix B from V you give me, I multiply it by the identity matrix I and the B I = B is also element of W.

2. When W contains a regular matrix it is quite similar - because then W must contain the identity matrix. Let A be a regular matrix from W, then if you give me an inverse matrix, when I multiply them then what I will get is the identity matrix and due to the aforesaid condition the identity matrix is an element of W.

But somehow I do not know how to continue.... Please help