Math 260, Fall 2017, Assignment 11

Algebra begins with the unknown and ends with the unknowable.

- Anonymous

Read:

1. Section 5.3.

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

1. Transpose (of a matrix).
2. Pattern (in an $n\times n$ matrix).
3. Symmetric matrix (you will need to look in the book; we did not discuss this in class).
4. Skew-symmetric matrix (as above).

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

1. Theorem characterizing when a matrix is orthogonal (in terms of orthonormality).
2. Theorem characterizing when a matrix is orthogonal (in terms of the transpose).
3. Formula for the inverse of an orthogonal matrix (in terms of the transpose).
4. Formulas for the transpose of a sum and the transpose of a product.
5. Theorem concerning orthogonality of the identity matrix, of the product of two orthogonal matrices, and of the inverse of an orthogonal matrix.
6. Formula for the number of patterns in an $n\times n$ matrix.

Solve the following problems:

1. Section 5.3, problems 1, 3, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 21, 23, and 25.
2. Section 6.1, problems 1, 2, 3, 4, 11, and 12.

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