Sherman–Morrison Formula

The Sherman-Morrison formula provides an efficient way to compute the inverse of a matrix after it has been updated by a rank-one perturbation. Specifically, if A is a nonsingular matrix and B = A + uv^T (where u and v are column vectors), the formula states:

Experiment with Matrix Updates

Open Sherman-Morrison Formula. We will observe what happens when the denominator is zero.

1. Click Reset. Start with the default Identity matrix: A11 = 1, A12 = 0, A21 = 0, A22 = 1.
2. Change the u vector to u1 = 1, u2 = 0.
3. Change the v vector to v1 = -1, v2 = 0.
4. Observe the Outer Product is now [[-1, 0], [0, 0]].
5. Matrix B becomes A + uv^T = [[0, 0], [0, 1]], which has a zero determinant (singular).
6. The Denominator (1 + v^TA⁻¹u) calculates to 0. The formula fails here, matching the fact that B is singular and has no inverse.