I have written a proof for this, but is it correct?

1. Is the set of all 3x3 SINGULAR matrices under matrix + and scalar * vector spaces?

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I define w={A3x3|det(A)=0}

I define A to be [ 1 1 1; 1 1 1; 1 1 1] and B [ 0 -1 -1; 0 0 -1;0 -1 0], both have determinant=0

but I notice that A + B = B+A is [1 0 0;1 1 0; 1 0 1] and det(a+b)=1

So, since the determinant is non0, that means that A+B is nonsingular. Since that violates the conditions of w, then the set of all 3x3 singular matricies is NOT A VECTOR SPACE

Is this sufficient proof for this?

Thanks!

Question

I have written a proof for this, but is it correct?

1. Is the set of all 3x3 SINGULAR matrices under matrix + and scalar * vector spaces?

__
I define w={A3x3|det(A)=0}

I define A to be [ 1 1 1; 1 1 1; 1 1 1] and B [ 0 -1 -1; 0 0 -1;0 -1 0], both have determinant=0

but I notice that A + B = B+A is [1 0 0;1 1 0; 1 0 1] and det(a+b)=1

So, since the determinant is non0, that means that A+B is nonsingular. Since that violates the conditions of w, then the set of all 3x3 singular matricies is NOT A VECTOR SPACE

Is this sufficient proof for this?

Thanks!

turingtest · Answer

sure looks that way to me

anonymous · Answer

Thanks! What can this be used for though (defining vector spaces and subspaces)? I am in a matrix theory class, and this seems to be quite different than what we have been doing, I'm not seeing any connection...