n and adj A M ij T ij T == == −− ++ In particular, adj A of a square matrix of order is given below: adj A M M M M M M M M M T == ++ −− ++ −− ++ −− ++ −− ++ == T == Theorem . For every square matrix A of order n , A A A A I n adj adj Proof For simplicity, we prove the theorem for n = only. Consider A . Then, we get a A a A a A a A a A a A a A a A a A a A a A a A a A a A ; a A a A a A a A a A a A a A ; a A a A a A a A a A a A A .
By using the above equations, we get adj AA A I … ( ) adj A A = aa A I … ( ) where I is the identity matrix of order . So, by equations ( ) and ( ), we get A A A A I adj adj Note If A is a singular matrix of order n , then