Exercises 12–17 develop properties of rank
Chapter , Problem 13E(choose chapter or problem)
Exercises 12–17 develop properties of rank that are sometimes needed in applications. Assume the matrix A is m × n.Show that if P is an invertible m × m matrix, then rank PA = rank A. [Hint: Apply Exercise 12 to PA and P–1 (PA)]Exercise 12:Show from parts (a) and (b) that rank AB cannot exceed the rank of A or the rank of B. (In general, the rank of a product of matrices cannot exceed the rank of any factor in the product.)a. Show that if B is n × p, then rank AB rank A. [Hint: Explain why every vector in the column space of AB is in the column space of A.]b. Show that if B is n × p, then rank AB rank B. [Hint: Use part (a) to study rank.(AB)T .]
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