A is a square matrix If columes of A are linearly independent, then the columes

Question

A is a square matrix If columes of A are linearly independent, then the columes ofA2 are also linearly independent T/F? Thanks!! A is a square matrix If columes of A are linearly independent, then the columes ofA2 are also linearly independent T/F? Thanks!!

Explanation / Answer

     True Proof :    Since A is a squarematrix , let it be of order n x n .    Now, as the columns of A are linearly independent , rank (A) = n   => A isinvertible or non-singular .    => there exists a matrix B of order n x n , the inverse of A , such that : BA = AB = I , I is theidentity matrix of order n x n    Now,   A2 = AA , is also a square matrix of order n x n .    and BBA2 = BBAA =B(BA)A = BIA =BA = I also ,    A2BB = AABB = A(AB)B=AIB =AB = I hence from above BB is the inverse of A2 or A2 is invertible    => rank (A2 ) = n   => the columns of A2 are linearly independent

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