Math 260, Fall 2018, Assignment 13

"Reeling and Writhing, of course, to begin with," the Mock Turtle replied; "And then the different branches of Arithmetic - Ambition, Distraction, Uglification, and Derision."

- Lewis Carroll, Alice's Adventures in Wonderland

Read:

1. Section 6.1.
2. Section 6.2.

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

1. Pattern (in an $n\times n$ matrix).
2. Inversion (in a pattern).
3. $NI(P)$ (the inversion count of the pattern $P$).
4. $\mathrm{prod}_P(M)$ (where $P$ is an $n\times n$ pattern and $M$ is an $n\times n$ matrix).
5. $\det(M)$.
6. Upper triangular (matrix).

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

1. Formula for the number of $n\times n$ patterns.
2. Explicit formula for the determinant of a $2\times 2$ matrix.
3. Explicit formula for the determinant of a $3\times 3$ matrix (a.k.a. "Sarrus' rule").
4. Warning concerning the erroneous extension of the pattern of Sarrus' rule to $4\times 4$ and larger matrices.
5. Formula for the determinant of an upper-triangular matrix.
6. Characteristic properties of the determinant.
7. Theorem concerning the determinant of a matrix with a repeated row.

Solve the following problems:

1. Section 6.1, problems 1, 3, 5, 7, 9, 11, 13, 15, 17, 31, and 39.
2. Section 6.2, problems 11, 13, 14, and 15.

--------------------End of assignment--------------------