Consider an n x n matrix A. Show that there existscalars c 0 , c 1 ,......., c n

Question

Consider an n x n matrix A. Show that there existscalars c0, c1,......., cn (notall zero) such that the matrixc0In + c1A +c2A2 + .....+ cnAn isnon-invertible. Hint: Pict an arbitrary nonzero vector v in Rn. Then then+1 vectors v, Av, A2v, ...., Anv will belinearly independent. (Much more is true: There are scalarsc0, c1,......., cn such that c0In + c1A +c2A2 + .....+ cnAn =0). Highest rating within 24hr. Thanks. Consider an n x n matrix A. Show that there existscalars c0, c1,......., cn (notall zero) such that the matrixc0In + c1A +c2A2 + .....+ cnAn isnon-invertible. Hint: Pict an arbitrary nonzero vector v in Rn. Then then+1 vectors v, Av, A2v, ...., Anv will belinearly independent. (Much more is true: There are scalarsc0, c1,......., cn such that c0In + c1A +c2A2 + .....+ cnAn =0). Highest rating within 24hr. Thanks.

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