Indicate whether the following statements are true or false

Indicate whether the following statements are true or false:

Ill-conditioning of equations occurs when the determinant of [A] is close to zero.

Solution

False.

The largeness of the condition number (A) measures the nearness to singularity, not the tininess of the determinant.

For example, the diagonal matrix 10^-50*(Identity matrix) has tiny determinant, but is well-conditioned.

On the other hand consider the upper triangular matrix

U = 1 2 2......2

0 1 2......2

0 0 1......2

................

0 0 0......1

Its determinant is always one but the ratio of the largest to the smallest singular value is shown to be cot^2(pi/4n) which can be seen to increase for increasing n.