topgear said:Homework Statement
Show that if A is an invertible upper triangular matrix, then A^-1 is also an upper triangular matrix
Homework Equations
The Attempt at a Solution
The inverse of something just flips the main diagonal and leaves everything else where is was just changing the signs. How do I say this in math language.
An upper triangular matrix is a square matrix in which all the elements below the main diagonal are zero. The main diagonal refers to the diagonal line from the top left corner of the matrix to the bottom right corner.
The inverse of an upper triangular matrix is another upper triangular matrix in which the elements on the main diagonal are the reciprocal of the corresponding elements in the original matrix, and the elements below the main diagonal are zero.
To find the inverse of an upper triangular matrix, you can use the Gauss-Jordan elimination method or the LU decomposition method. Both methods involve performing row operations on the original matrix to transform it into an identity matrix, with the inverse matrix appearing on the other side of the identity matrix.
Finding the inverse of an upper triangular matrix can be useful in solving systems of linear equations, as well as in other applications such as calculating determinants and solving differential equations.
Some properties of the inverse of an upper triangular matrix include: it is also an upper triangular matrix, the product of the original matrix and its inverse is the identity matrix, and the inverse of the inverse matrix is the original matrix.