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
$\displaystyle A^{-1}=\frac{1}{det(A)}\cdot adj(A)$
We can prove that $\displaystyle A^{-1}$ 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 $\displaystyle C_{ij}$ with
i<j(above the diagonal) is 0.
Since $\displaystyle C_{ij}=(-1)^{i+j}M_{ij}$ it suffices to show that
each minor $\displaystyle M_{ij}$ with i<j is 0.
Start by letting $\displaystyle B_{ij}$ 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 $\displaystyle det(A)=a_{11}a_{22}.....a_{nn}$