Question

4. Let A and B be two nx n matrices. Suppose that AB is invertible. Show that the system A.x = 0 has only the trivial solution.

Answer 1

Since AB is invertible that is determinant of AB exists. That is we can have (AB)^(-1)

Since,

(AB)^(-1)=B^(-1) A^(-1)

Which is only possible when A^(-1) and B^(-1) both exists.

So A is invertible. That is rank of A is n ,since A is invertible.

Therefore, by Rank-Nullity theorem if dim A=n

Then,

Dim(A)=RANK(A)+ NULLITY(A).

So ,

n=n+NULLITY(A).

=> NULLITY(A)=0.

And Nullity is {x|Ax=0}

So, nullity of A has only trivial solution.