九州大学 システム情報科学府 情報理工学専攻・電気電子工学専攻 2024年8月実施 線形代数

Author

祭音Myyura (assisted by ChatGPT 5.5 Thinking)

Description

(1) Find the determinants of the following matrices and , respectively.

(2) For any upper triangular matrix of order , show

where denotes the determinant of .

Kai

(1)

First compute .

Expanding along the first row, we get

Thus,

Therefore,

Hence,

Next compute .

Notice that the third row is twice the first row:

Therefore, the rows of are linearly dependent, so

Hence,

(2)

Let be an upper triangular matrix of order . By the Leibniz formula for determinants,

Since is upper triangular, we have

Therefore, for the product

to be nonzero, it is necessary that

However, since is a permutation of , we have

Thus, if for all , then we must have

Hence, the only nonzero term in the Leibniz formula comes from the identity permutation.Therefore,

Hence,