Revision as of 06:02, 28 April 2011

MA 265 Chapter 3 Sections 3.1-3.2

By: Daniel Ford

What are determinants?

To understand determinants, you must first know about permutations.

Permutations

If D = {1, 2,....,n} a set of integers from 1 to n in ascending order, then a permutation is the rearrangement of an integer in D.

Example:

If D = {1, 2, 3, 4}, then 2134 would be a permutation of D.  This corresponds to the function f: D→ D defined by

        σ(1) = 1
        σ(2) = 2
        σ(3) = 3
        σ(4) = 4
Then after permutation,
        σ(1) = 2
        σ(2) = 1
        σ(3) = 3
        σ(4) = 4

The reason that this is important to know is that the total number of permutations can be even or odd, depending on the number of inversions.  So in the permutation 4321, where 4 precedes 3, 4 precedes 1, 4 prececedes 2, 3 precedes 1, 3 precedes 2, and 2 precedes 1.  Thus the total number of inversions in this premutation is 6, which would make 4321, even.  This determines what sign you put in front, so if it is odd, then you put a negative sign (-) in front; and if it is even, then you put a postive sign (+) in front.

Determinants

Now that you know what permutations are, we can start to talk about determinants.

Let A = [a_ij ] be an n x n matrix.  The determinant function, denoted by det, is defined by

                    det(A) = ∑(±)a_1j1a_2j2•••a_njn

where the summation is over all permutations j₁j₂•••j_n of the set D = {1,2,...,n}.  The sign is taken as positive (+) or negative (-) according to whether the permutation j₁j₂•••j_n is ever or odd.

Example:

If A = [a₁₁] is a 1 x 1 matrix, then det(A) = a₁₁.

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)

Example:

When you need to compute the determinant of a 4 x 4 matrix or greater, you can not use the cross-multiply and add/subtract formula that you used to compute that of a 3 x 3 matrix or lower.  So in order to compute the determinant you must row reduce the matrix, shown below.

Steps:

1. Multiply row 1 by -1/2 and add to row 2.
2. Multiply row 1 by -3/4 and add to row 3.
3. Multiply row 2 by -3/2 and add to row 3.
4. Interchange rows 3 and 4. (**IMPORTANT NOTE** Row replacement does not change det)

  $  A =         \begin{bmatrix}         4 & 2 & 2 & 0\\         2 & 0 & 0 & 0\\         3 & 0 & 0 & 1\\         0 & 0 & 1 & 0\end{bmatrix} $  $    =         \begin{bmatrix}         4 & 2 & 2 & 0\\         0 & -1 & -1 & 0\\         0 & -3/2 & -3/2 & 1\\         0 & 0 & 1 & 0\end{bmatrix} $  $    =         \begin{bmatrix}         4 & 2 & 2 & 0\\         0 & -1 & -1 & 0\\         0 & 0 & 0 & 1\\         0 & 0 & 1 & 0\end{bmatrix} $  $    = (-1)         \begin{bmatrix}         4 & 2 & 2 & 0\\         0 & -1 & -1 & 0\\         0 & 0 & 1 & 0\\         0 & 0 & 0 & 1\end{bmatrix} $

After you have row reduce the matrix you can multiply the diagonal of the matrix because the matrix is in in triangular form (which means there is a triangle of zeros in the bottom or upper part of the matrix).  **One thing to note is that when you do a row interchange, during the row reduction process, make sure to put a (-1) out in front of the matrix, but this will not change the value of the determinant.** So your final answer is:

                                    det(A) = (-1)(4)(-1)(1)(1) = 4

Properties of Determinants

There are some different properties of determinants that are helpful when computing determinants.

Theorems:

• If A is a matrix, then det(A) = det(A^T).
• If matrix B results from matrix A by interchanging two different rows (columns) of A, then det(B) = –det(A).
• If two rows (columns) of A are equal, then det(A) = 0.
• If a row (column) of A consists entirely of zeros, then det(A) = 0.
• If B is obtained from A by multiplying a row (column) of A by a real number k, then det(B) = k det(A).
• If A is an n x n matrix, then A is nonsingular if and only if det(A) ≠ 0.
• If A and B are n x n matrices, then det(AB) = det(A)det(B).

Corollaries:

• If A is an n x n matrix, then Ax = 0 has a nontrivial solution if and only if det(A) = 0.
• If A is nonsingular, then det(A^-1) = 1/det(A).
• If A and B are similar matrices, then det(A) = det(B). 

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