Let A be an m╳n real matrix, and let A' be the transpose of A.
(a) Show that if m = 2 and n = 4, then the determinant of A'A = 0.
(b) Write down certain conditions on A, m and n which will ensure
that the determinant of A'A is nonzero.
(c) Show that if v_1, v_2, ..., v_n are linearly independent
vectors of R^n, then the determinant of
[ (v_1,v_1) (v_1,v_2) ... (v_1,v_n) ]
[ (v_2,v_1) (v_2,v_2) ... (v_2,v_n) ]
[ ......... ......... ... ......... ]
[ (v_n,v_1) (v_n,v_2) ... (v_n,v_n) ]
is positive, where ( , ) is the standard inner product of R^n.
(d) Is there any relationship between the rank of A'A and the
rank of A ?
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