Can you help me with the following question please
Prove that the inverse of a non singular N X N upper triangular matrix is an upper triangular matrix
We can assume that the matrix A is upper triangular and invertible, since
We can prove that is upper triangular by showing that
the adjoint is upper triangular or that the matrix of cofactors is lower
triangular. Do this by showing every cofactor with
i<j(above the diagonal) is 0.
Since it suffices to show that
each minor with i<j is 0.
Start by letting be the matrix we get when the ith row
and jth column of A are gotten rid of (deleted).
Can you go further?.
There is a handy theorem which we won't prove, but use, which says:
"If A is an nXn triangular matrix then det(A) is the product of the entries on the main diagonal of the matrix. What I mean is