Does the basis of column space of a square matrix remains the same after squaring i.e is c(A)=c(AxA)?

Well I'm working on matrices, and this thing is bugging me , though according to me it should be true because: let there be a nxn matrix:(aij ; i=row number, j=column number)

a11 a12 a13.....a1n
a21 a22 a23......
a31 a32 a33......
.   .   .  ......
.   .   .  ......
an1 an2 an3......

then A^2 will be:

1             column 1                          column 2
a11*a11+a12*a21+a13*a31....+a1n*an1|a11*a12+a12*a22+a13*a32....+a1n*an2|
a21*a11+a22*a21+a23*a31....+an2*an1|a21*a12+a22*a22+a23*a32....+an2*an2|

.                                  |.                                  |
.                                  |.                                  |
.            开发者_JS百科                      |.                                  |
an1*a11+an2*a21+an3*a31....+ann*an1|an1*a12+an2*a22+an3*a32....+ann*an2|

so if we take the row elements to be constants x,y,z,..... then all the columns are essentially x*column1+y*column2............ What do you all think?

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No. Consider the matrix [0,1;0,0] (using matlab notation). Its square is [0,0;0,0]. That doesn't have the same column space.