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Prove the two invertible matrices U,V exist

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December 8, 2010
linear algebramatrixvectorinductionlinear algebra unsolved

Problem Statement

Let A∈M4(C)A\in M_4(C) be a non-zero matrix. a)a) If rank(A)=r<4\text{rank}(A)=r<4, prove the existence of two invertible matrices U,V∈M4(C)U,V\in M_4(C), such that: UAV=(Ir000)UAV=\begin{pmatrix}I_r&0\\0&0\end{pmatrix} where IrI_r is the rr-unit matrix. b)b) Show that if AA and A2A^2 have the same rank kk, then the matrix AnA^n has rank kk, for any n≥3n\ge 3.
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