Let A be an n × n matrix with det A = 1 and with all elements of A integers.
(a) Show that A−1 has only integer entries.
(b) Suppose that b is an n-vector with only integer entries. Show that the solution vector x of Ax = b has only integer entries.
Let A be a 3 × 3 upper triangular matrix with nonzero determinant. Show by explicit computation that A−1 is also upper triangular.
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