Math 260, Fall 2018, Assignment 14

Algebra begins with the unknown and ends with the unknowable.

- Anonymous

Read:

1. Section 7.1.

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

1. Eigenvector (of a matrix).
2. Eigenvalue (associated with an eigenvector).

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

1. Theorem concerning the effect of elementary row operations on determinants.
2. Theorem relating determinants to invertibility.
3. Theorem concerning Laplace expansion across rows.
4. Theorem concerning Laplace expansion down columns.

Carefully describe the following algorithms:

1. Algorithm to compute determinants by row reduction.

Solve the following problems:

1. Section 6.2, problems 1, 3, 5, 7, 9, and 45. (Hint for problem 45: use Laplace expansion across the fourth row. Four of the five terms are irrelevant to the question.)
2. Compute the determinant of the following matrix: $$A=\begin{bmatrix}5&4&0&0&0\\6&7&0&0&0\\3&4&5&6&7\\2&1&0&1&2\\2&1&0&0&1\end{bmatrix}.$$ (Hint: start with Laplace expansion down the third column.)
3. Section 7.1, problems 1, 3, 5, 7, 8, and 9.

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