Measure the test taker's knowledge about Matrix Rank 1 disturbance.

This is a common linear algebra problem:

 

Example:Evaluate the determinant of $ N $ level matrix $ A $, where \ [A _ {IJ} =\left \ {\ begin {array} {L} X & I = J \ y & I <J \ Z & I> J \ end {array} \ right. \ quad. \]

 

At first glance, this matrix is a bit complex, but its law is also obvious.

The method we want to introduce is called "rank 1 disturbance". It uses the simple fact that $ A and B $ are two square arrays, and the rank of $ B $ is 1, then $ F (t) = \ det | A + Tb | $ is a polynomial of $ T $.

In this example, if $ F (t) = \ det (A _ {IJ} + T) $, then $ F (t) $ is a polynomial of $ T $ and $ F (-y) = (x-y) ^ N $, $ F (-z) = (X-Z) ^ N $, use these two values to calculate the constant term $ F (t) $ F (0) = \ det A $.

When solving the determinant, we introduce variables to regard the final determinant as one or more variable functions, finding out the relationships that this function may satisfy (differential equations, recursive relationships, root, etc.) is an important idea.

Measure the test taker's knowledge about Matrix Rank 1 disturbance.