Bonus point project for MA265.

# MA 265 Chapter 3 Sections 3.1-3.2: A Review

# By: James Jacob

**Section 3.1: Defining Determinants**

Determinants are not as efficient as methods for solving systems like in Chapter 2. Determinants are also important in linear transformations when discussed in Chapter 6.

First there are **permutations**. If P = {1,2,.....,n} a set of integers from 1 to n in ascending order, then a permutation would be every rearrangement of an integer in P.

**Example:**

For example if P = {4,5,6,7}, then 5467 would be a permutation of P. First,

f(1) = 4 f(2) = 5 f(3) = 6 f(4) = 7

Then after permutation,

f(1) = 5 f(2) = 4 f(3) = 6 f(4) = 7

So any element in P can be in any position and each "new" set using the same elements is a permutation. The number of total permutations a set can have can be determined by the number n elements. The total number of permutations is equal to n! (*n* factorial).

Permutations can have **inversions** if a larger integer comes before a smaller one in the set. If the total number of inversions is even, the permutation is **even**. If the total number of inversions is odd, then the permutation is **odd**.

**Example:**

If a permutation = 6543, 6 is larger and comes before 5, 4, and 3, 5 comes before 4 and 3, and 4 comes before 3 which totals to 6 inversions therefore the permutation is even. Now if the permutation = 6345, 6 comes before 3, 4, and 5 which totals to 3 inversions therefore the permutation is odd. If the number of elements *n* is greater than or equal to two in a set then there are *n*!/2 even and *n*!/2 odd functions.

Determinant is also written as **det** and is defined as the summation of all permutations of a set A. Each permutation is positive or negative depending on whether it is even or odd respectively. Each term of determinant set A is a product of *n* entries, so one entry from each row and one from each column. There are *n*! terms in the sum.

**Example:**

If

```
$ A = \begin{bmatrix} a1 & a2\\ a3 & a4 \end{bmatrix} $
```

then

det(A) = a1a4 - a2a3

**Example:**

If

```
$ A = \begin{bmatrix} a1 & a2 & a3\\ a4 & a5 & a6\\ a7 & a8 & a9 \end{bmatrix} $
```

then

det(A) = (a1a5a9 + a2a6a7 + a3a4a8) - (a3a5a7 + a2a4a9 + a1a6a8)

**Section 3.2: Properties of Determinants**

- Theorem 3.1: If A is a matrix, then the determinant of A is equal to the determinant of A^T

det(A) = det(A^T)

- Theorem 3.2: A matrix B can result from matrix A, if two different rows or columns are interchanged within A

det(B) = -det(A)

- Theorem 3.3: In the matrix A if two rows or columns are the same then the determinant of A equals zero
- Theorem 3.4: In the matrix A is there is a row or column of zeros then the determinant of A equals zero
- Theorem 3.5: If the matrix A is equal to matrix B by multiplying a row or column of A by a real number
*k*then determinant of B is equal to determinant of A times*k*

det(B) =kdet(A)

- Theorem 3.6: If any elementary option is used on matrix A to get matrix B then determinant of A is equal to determinant B
- Theorem 3.7: If matrix A is an upper or lower triangular then the determinant of A is the product the values on the main diagonal
- Theorem 3.8: If A is an
*n*x*n*then A is nonsingular if and only if determinant of A does not equal zero - Theorem 3.9: If A is an
*n*x*n*matrix and so is B then the determinant of A time B is equal to the determinant A times the determinant of B

det(AB) = det(A)det(B)

- Corollary 3.1: If determinant of A, an
*n*x*n*matrix, is equal to zero then A**x**=**0**has a nontrivial solution - Corollary 3.2: If A is nonsingular then the determinant of inverse of A is equal to one divided by the determinant of A

det(A^-1) = 1/det(A)

- Corollary 3.3: If matrix A and matrix B are similar then the determinant of A is equal to determinant of B