Let A be a lower upper triangular nonsingular matrix Show th

Let A be a lower (upper) triangular nonsingular matrix. Show that A-1 is also lower (upper) triangular. If A and B are (n Times n) nonsingular matrices, show that AB and BA are also nonsingular. Furthermore, show that (AB)-1 =B-1A-1.

Solution

Let us prove for lower triangular matrix and the same holds for upper triangular also.

A is given to be lower triangular matrix.

i.e. A has zero elements for aij whenever i>j

As product of two lower triangular only will give a lower triangular

and as I is also lower triangular, we have

AA inverse = I implies

A inverse must be lower triangular if A is lower triangular

Hence proved.

12) det(AB)=detAdetB

Hence when A and B are non singular

det AB cannot be 0 and hence non singular.

A is non singular hence A^-1 exists, similarly B^-1 exists.

Consider B^-1A^_1(AB) = B^-1(A^_1A)B

= B^-1IB

= I

Hence proved.