Let A be an n × n matrix and denote the (k, j)-cofactor of A. (a) Prove that if P is...

Let A be an n × n matrix and denote the (k, j)-cofactor of A.

(a) Prove that if P is the matrix obtained from A by replacing column k by then det P = Show that for each j, we have

Hint: Apply Cramer's rule to Ax =

(c) Deduce that if B is the n × n matrix whose (i, j)-entry is , then AB = This matrix B is called the classical adjoins of A.

(d) Show that if det A ≠ 0,