Invertibility of a product of invertible matrices

[tobias_funke]

If A,B, and C are each nxn invertible matrices, will the product ABC be invertible?

 

[jakncoke]

yea, if A, B are nxn invertible then AB is invertible. Since Matrix Multiplication is associative, (AB)C is invertible provided c is nxn invertible.

 

[Landau]

If your linear algebra is better than your basic set/function theory, remark that A,B,C have non-zero determinant, hence their product also.

 

[HallsofIvy]

If A and B are both invertible then (AB)(B^{-1}A^{-1})= A(B^{-1}B)A^{-1} because, as jakncoke says, matrix multiplication is associative. Of course, B^{-1}B= I so that becomes (AB)(B^{-1}A^{-1})= AA^{-1}= I. Simlarly, (B^{-1}A^{-1})(AB)= B^{-1}(A^{-1}A)B^{-1}= B^{1}B= I

You can extend that to any number of factors by induction and repeated use of associativity.

 

[mathwonk]

As Landau referenced, if you like to think about functions, matrix multiplication is equivalent to applying a linear map, and invertible matrices define bijective linear maps (and vice versa), and a composition of bijective linear maps is a bijective linear map. so yes.