Question

For square matrices A and B of the same order, we have adj(AB) = ? 

A. (adj A)(adj B) 

B. (adj B)(adj A) 

C. │AB│ 

D. none of these

Answer 1

We know that (AB)^-1 = adj(AB)/ |AB|

adj (AB)= (AB)^-1 |AB|

We also know that (AB)^-1 = B^-1. A^-1 

|AB| = |A| |B|

Putting them in 1 

Adj (AB) = B^-1. A^-1. |A|.|B|

= (A^-1. |A|) (B^-1 |B|) 

= adj(A) adj(B) 

Since, adj (A)= (A)^-1 |A|

adj (B)= (B)^-1|B|