# 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 Ax = 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