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Unformatted text preview: 4. A square matrix A is called a nilpotent if A r = 0 for some natural number r . (a) Let A = ( a ij ) n × n be a matrix whose entries satisfy a ij = 0 if i ≥ j . Show that A is a nilpotent. (b) Is there a nilpotent matrix such that all of its entries are nonzero? Justify your answer. 1 5. Let A = ± 1 2-1 1 ² and C = ±-1 1 2 1 ² . (a) Find elementary matrices E 1 and E 2 such that C = E 2 E 1 A . (b) Show that there is no elementary matrix E such that C = EA . 6. In each case ﬁnd an invertible matrix U such that UA = R in reduced row-echleon form and express U as a product of elementary matrices. (a) A = ± 1 2 1 5 12-1 ² . (b) A = 2 1-1 0 3-1 2 1 1-2 3 1 . 7. Prove that B is row equivalent to A if and only if there exists a nonsingular matrix M such that B = MA . 8. ( Harder ) Let A be an m × n matrix, and B be n × m . Prove that if n < m , then AB is not invertible. 2...
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