prove inverse of upper triangular matrix

**flaming** · October 25th 2008, 02:26 PM

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

**galactus** · October 25th 2008, 02:43 PM

We can assume that the matrix A is upper triangular and invertible, since

$A^{-1}=\frac{1}{det(A)}\cdot adj(A)$

We can prove that $A^{-1}$ 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 $C_{ij}$ with

i<j(above the diagonal) is 0.

Since $C_{ij}=(-1)^{i+j}M_{ij}$ it suffices to show that

each minor $M_{ij}$ with i<j is 0.

Start by letting $B_{ij}$ 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 $det(A)=a_{11}a_{22}.....a_{nn}$

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