Homework Chapter 2 UTK – M251 – Matrix Algebra Fall 2003, Jochen Denzler, MWF 1:25–2:15, Ayres 318

1. How many inversions does the permutation (3, 1, 5, 2, 7, 4, 6) have? (In other words, we are taking about the permutation that gives the term a_{13}a_{21}a_{35}a_{42}a_{57}a_{64}a_{76} in the determinant of a 7 × 7 matrix) – Is it an even or an odd permutation?

2. Read the glossary entry on even and odd permutations. Find a sequence of swaps of pairs of numbers that obtains (3, 1, 5, 2, 7, 4, 6) from (1, 2, 3, 4, 5, 6, 7). How many swaps did you take?

3. How many inversions does the permutation (n, n − 1, n − 2, …, 3, 2, 1) have? List the number of inversions for the cases n = 2, 3, 4, …, 9 in a table.

4. Look at ex. 8 on p. 86 and make sure that you understand the special rule memorization

device for 3 × 3 determinants. Use it to calculate | \begin{matrix}2 & −1 & 3 \\ 4 & 1 & −2 \\ 1 & 1 & 5 \end{matrix} |. It’s also a good idea

to try to do this same technique without actually copying the two columns on paper.

5. Complete the following sentence: If two columns (or two rows) in a square matrix are equal, then the determinant of this matrix . . . . . . . . . . . . . . . . .

6. Calculate the following determinant by the use of row transformations:

| \begin{matrix}2 & −1 & 0 & 0 & 0 & 0 \\ −1 & 2 & −1 & 0 & 0 & 0 \\ 0 & −1 & 2 & −1 & 0 & 0 \\ 0 & 0 & −1 & 2 & −1 & 0 \\ 0 & 0 & 0 & −1 & 2 & −1 \\ 0 & 0 & 0 & 0 & −1 & 2 \end{matrix} |

7. Have a close look at your work in the previous problem. You should discern a pattern. I give you a 100 × 100 matrix with the same pattern, namely 2’s on the diagonal and −1’s next to them, 0’s everywhere else. What is the determinant of this matrix?

8. Calculate

det [ \begin{matrix}3 & 5 & 1 & 2 \\ 1 & 10 & 1 & 2 \\ 0 & −5 & 1 & 2 \\ 4 & 15 & 2 & 1 \end{matrix} ]

9. Find the following determinant, using a smart sequence of row or column transformations to get many zeros easily:

| \begin{matrix}1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 2 & 2 & 2 & 2 & 2 & 2 & 1 \\ 3 & 3 & 3 & 3 & 3 & 2 & 1 \\ 4 & 4 & 4 & 4 & 3 & 2 & 1 \\ 5 & 5 & 5 & 4 & 3 & 2 & 1 \\ 6 & 6 & 5 & 4 & 3 & 2 & 1 \\ 7 & 6 & 5 & 4 & 3 & 2 & 1 \end{matrix} |

Hint: There are at least two choices: create zeros in the first column below a leading one in the first row, then cofactor expansion of the first column. Or: bottom-up row operations: for i = n, n − 1, … 2, subtract the (i − 1)^{st} row from the i^{th} row.

10. Let

V := | \begin{matrix}1 & 1 & 1 \\ x & y & z \\ x^{2} & y^{2} & z^{2} \end{matrix} |

Before you start calculating, answer the question: What value does V have, if x = y? — Now calculate V for general x, y, z. Here comes the punchline: I want you to write the result not as a mess with 6 terms, but transform it into a product (∗ − ∗)(∗ − ∗)(∗ − ∗), where you have to figure out what the ∗ stand for. And the very first question in this problem will serve as a hint how to accomplish this task.

11. In the following matrix the symbols e and o stand for even and odd integers respectively. I don’t reveal for which exactly. Different e’s may stand for different even numbers, and similarly for o’s. I claim that from this information you can already know that the determinant

| \begin{matrix}o & e & e & e & e \\ e & o & e & e & e \\ e & e & o & e & e \\ e & e & e & o & e \\ e & e & e & e & o \end{matrix} |

is not zero. Explain how you can come to this conclusion.

12. For which numbers a does the linear system

ax + 2y − 3z = 5
2x + 3y − 2z = 7
3x + 2y − 5z = 9

have exactly one solution? Use Cramer’s rule to find y in this case. (Finding x and z is not required.)

13. For A as given below, find adj(A).

A = [ \begin{matrix}1 & −1 & 2 \\ −2 & a & −3 \\ 0 & −2 & a \end{matrix} ]

14. Let A be a 5 × 5 matrix with determinant −2. Simplify adj(adj(A)) to an expression that can be typed on the computer keyboard with 5 or less keystrokes. Make sure that you give a clear calculation to justify your result. Hint: If you find this difficult, you should collect formulas about the adjoint of a matrix and the determinant of matrices from the glossary, and you need to look which are useful and try to put them together. Quote the formulas you are using.

15. If all entries of a square matrix A are integers and if furthermore det A = 1, can you conclude that all entries of A^{−1} are integers? If the answer is yes, explain why; if the answer is no, explain why not (counterexample).