Efficient Calculation of Determinant on change for 3X3 matrix?

Question

Let's say you were given a 3x3 matrix of values and only a particular column changes values between iterations.

On Step 1 I have X:

[2.0, 4.6, 3.4]
[3.2, 6.7, 4.1]
[2.1, 1.4, 5.3]

Whose determinant is -11.476

Now on Step 2 I have X, with the second column populated with new values.

[2.0, 6.5, 3.4]
[3.2, 3.4, 4.1]
[2.1, 0.8, 5.3]

Is there a quick way to calculate the determinant of this matrix given the previous state of the matrix and its previous determinant? I want to preserve some of the information known at the previous state. Only columns change on each iteration.

Answer 1

如果更改的列始终相同，则可以对该列使用Laplace扩展 。

Answer 2

If your first and third columns are constant and only second column varies, then you can transform this into a formula:

[2.0, a, 3.4]
[3.2, b, 4.1]
[2.1, c, 5.3]

= -ax [3.2, 4.1][2.1, 5.3] + bx [2.0, 3.4][2.1, 5.3] - cx [2.0, 3.4][3.2, 4.1]

= -8.35 xa + 3.46 xb + 2.68 xc

So now you have a formula that you can use.