Math 260, Fall 2019, Assignment 12

"When I think of Euclid even now, I have to wipe my sweaty brow."

- C. M. Bellman

Read:

1. Section 6.1.

Carefully define the following terms, then give one example and one non-example of each:

1. $n\times n$ pattern.
2. $\mathrm{prod}_P(A)$ (where $P$ is an $n\times n$ pattern and $A$ is an $n\times n$ matrix).
3. Inversion (in a pattern $P$).
4. $\mathrm{inv}(P)$ (the inversion count of a pattern $P$).
5. $\mathrm{det}(A)$ (the determinant of the $n\times n$ matrix $A$).

Carefully state the following theorems (you do not need to prove them):

1. Formula for the determinant of a $2\times 2$ matrix.
2. Sarrus' Rule (for the determinant of a $3\times 3$ matrix).
3. Formula for the number of $n\times n$ patterns.
4. Warning concerning the misuse of Sarrus' Rule in sizes other than $3\times3$.

Solve the following problems:

1. Section 6.1, problems 1, 3, 5, 7, 9, 11, 13, 43, 45, and 54.

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