Let A be an m x n matrix. Let C = (cij) be the m x m matrix which differs from the identity only in that chh= λ ≠ 0. Let B be the m x m matrix which differs from the identity only in the kth column, in which bkk = 1.
a) Show that CA is obtained from A by multiplying the h th row by λ.
b) Show that BA is obtained from A by adding bjk times the kth row to the jth row for all j except k.
c) Show that B and C are nonsingular
Remark These results show that steps I and III of Section 1.10 are obtained by multiplying . left by a nonsingular matrix. The same assertion applies to step II (Problem 11 following Section 1.9).
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