Let A be the set of all n x n matrices with det 1 Is A close

Let A be the set of all n x n matrices with det 1. Is A closed under multiplication? Prove or disprove.

Solution

Let A be the set of all n x n matrices with det 1

take X,Y in A

then det(X)=1 and det(Y)=1

Since det(XY)=det(X)det(Y)

A closed under multiplication

Therefore, det(XY) = 1.1

                            = 1

Hence XY is in A

therefore A closed under multiplication