Math 260, Fall 2017, 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 6.2.
2. Section 6.3.

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

1. $A_{i,j}$ (i.e. the $i,j$ minor of a matrix $A$).

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

1. Theorem concerning the linearity of the determinant in individual rows.
2. Theorem concerning the effect of row swaps on determinants.
3. Theorem concerning the determinant of a matrix with a repeated row.
4. Theorem concerning the determinant of a matrix with a row of zeros.
5. Theorem concerning the determinant of the identity matrix.
6. Theorem concerning the effect of elementary row operations on the determinant.
7. Theorem relating determinants to invertibility.
8. Laplace expansion theorem.
9. Theorem concerning the determinant of the transpose.
10. Theorem concerning the determinant of a product.
11. Theorem concerning the determinant of an orthogonal matrix.
12. Theorem concerning the hypervolume of the hyperparallelepiped determined by the columns of a matrix.

Carefully describe the following algorithms:

1. Procedure to compute the determinant by row-reduction.

Solve the following problems:

1. Section 6.2, problems 1, 3, 5, 7, 9, 11, 13, and 15.
2. Section 6.3, problems 1, 3, 9, and 13 (see Theorem 6.3.6).

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