Prove that if U is an invertible upper triangular matrix then U^-1 is also upper triangular. How would you prove this?????? TIA

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- Oct 14th 2006, 09:40 AMTreeMoneyInvertible Upper Triangular Matrix
Prove that if U is an invertible upper triangular matrix then U^-1 is also upper triangular. How would you prove this?????? TIA

- Oct 14th 2006, 10:18 AMCaptainBlack
Consider inverting U using Gaussian elimination. We form the augmented

matrix

U|I,

which is obtained by appending the identity matrix on the right of U.

Then using row operations we reduce the left half of this augmented

matrix to the identity matrix, and we are left with the inverse of U in

the right half of the augmented matrix.

Now if we start from the bottom working our way up with the elimination

process the right half of the augmented matrix remains upper triangular, hence

when we finish with the identity in the left half of the augmented matrix

we have an upper triangular matrix in the right half of the augmented

matrix. But this upper triangular is the inverse of U, which proves that

the inverse is upper triangular.

RonL - Oct 14th 2006, 02:00 PMTreeMoney
Thanks RonL. I had no chance of coming up with something like that. Much Appreciated. John

- Oct 14th 2006, 02:02 PMCaptainBlack