Explain why if Ax rightarrow 0 has a solution x rightarrow

Explain why if Ax rightarrow = 0 has a solution x rightarrow = (4 5 6), then Ax rightarrow = 0 has infinitely many solutions. If A, B, C are invertible n times n matrices then show that AB^-1 C is also invertible.

Solution

Let, c be any real number

A(cx)=c(Ax)=c*0=0

Hence, cx is also a solution for any real number c

HEnce there are infinitely many solutions to Ax=0

det(A),det(B),det(C) are all non zero since, A,B,C are not invertible

HEnce, det(B^{-1})=1/det(B) is non zero

HEnce, det(AB^{-1}C)=det(A)det(B^{-1})det(C) = non zero

Hence, AB^{-1}C is invertible