Let A be an n x n matrix whose only nonzero entries are in the first...

 

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Let A be an n x n matrix whose only nonzero entries are in the first column and first row (i.e., aij = 0 when i > 1 and j > 1). (a) Show that A is of rank ≤ 2. When is the rank less than 2? (b) Assume that in addition A is symmetric and that a11 = 1. Show that there exist two vectors u and v such that A = uur - vvT. Let A be an n x n matrix whose only nonzero entries are in the first column and first row (i.e., aij = 0 when i > 1 and j > 1). (a) Show that A is of rank ≤ 2. When is the rank less than 2? (b) Assume that in addition A is symmetric and that a11 = 1. Show that there exist two vectors u and v such that A = uur - vvT.