Numerical Linear Algebra Question

[bjgolub]

Hello all,

Here is a question Ive been struggling with for a few days. Its more theoretical than computational, which I think I am having so much trouble with it.

Suppose A is an nxn matrix, and is nonsingular.
Assume for a particular i and j that there is no way to make A singular by changing the value of a_ij. What can be concluded about the A^-1?
Hint: Use the Sherman-Morrison Formula:

Suppose \(\displaystyle A\) is an invertible square matrix and \(\displaystyle u,\, v\) are vectors. Suppose furthermore that \(\displaystyle 1\, +\, v^T\, A^{-1}\, u\, \neq\, 0\). Then the Sherman–Morrison formula states that

. . . . .\(\displaystyle \left(A\, +\, uv^T\right)^{-1}\, =\, A^{-1}\, -\, \dfrac{A^{-1}\, uv^T\, A^{-1}}{1\, +\, v^T\, A^{-1}u}\)

Here, \(\displaystyle uv^T\) is the outer product of two vectors \(\displaystyle u\) and \(\displaystyle v\).

Any help would be greatly appreciated.

Thanks!