Review of Determinants

Rabu, 25 Desember 2019

In this section we are going to do a quick review of determinants and we’ll be concentrating almost exclusively on how to compute them. For a more in depth look at determinants you should check out the second chapter which is devoted to determinants and their properties. Also, we’ll acknowledge that the examples in this section are all examples that were worked in the second chapter.

We’ll start off with a quick “working” definition of a determinant. See The Determinant Function from the second chapter for the exact definition of a determinant. What we’re going to give here will be sufficient for what we’re going to be doing in this chapter.

So, given a square matrix, A, the determinant of A, denoted by det ( A), is a function that associated with A a number. That’s it. That’s what a determinant does. It takes a matrix and associates a number with that matrix. There is also some alternate notation that we should acknowledge because we’ll be using it quite a bit. The alternate notation is, det ( A) = A .

We now need to discuss how to compute determinants. There are many ways of computing determinants, but most of the general methods can lead to some fairly long computations. We will see one general method towards the end of this section, but there are some nice quick formulas that can help with some special cases so we’ll start with those. We’ll be working mostly with matrices in this chapter that fit into these special cases.

We will start with the formulas for 2× 2 and 3×3 matrices.

Definition 1 Ifthen the determinan of A is,

                                               

= a₁₁ a₂₂ – a₁₂ a₂₁

Definition 2 Ifthen the determinan of A is,

           

 

                 = a₁₁ a₂₂ a₃₃ + a₁₂ a₂₃ a₃₁ + a₁₃ a₂₁ a₃₂ – a₁₂ a₂₁ a₃₃ –  a₁₁ a₂₃ a₃₂  – a₁₃ a₂₂ a₃₁

Okay, we said that these were “nice” and “quick” formulas and the formula for the 2×2 matrix is fairly nice and quick, but the formula for the 3×3 matrix is neither nice nor quick. Luckily there are some nice little “tricks” that can help us to write down both formulas.

As we can see from this example the determinant for a matrix can be positive, negative or zero.

Likewise, as we will see towards the end of this review we are going to be especially interested in when the determinant of a matrix is zero. Because of this we have the following definition.

Definition 3 Suppose A is a square matrix.

(a)   If det (A) = 0 we call A a singular matrix.

(b)   If det (A) ≠ 0 we call A a non-singular matrix.

So, in Example 1 above, both A and B are non-singular while C is singular.

Before we proceed we should point out that while there are formulas for larger matrices (see the Determinant Function section for details on how to write them down) there are not any easy tricks with diagonals to write them down as we had for 2×2 and 3×3 matrices.

With the statement above made we should note that there is a simple formula for general matrices of certain kinds. The following theorem gives this formula.

Theorem 1 Suppose that A is an n × n triangular matrix with diagonal entries a₁₁ , a₂₂ , … , a_nn the determinant of A is,

det (A) = a₁₁ , a₂₂ , … , a_nn

This theorem will be valid regardless of whether the triangular matrix is an upper triangular matrix or a lower triangular matrix. Also, because a diagonal matrix can also be considered to be a triangular matrix Theorem 1 is also valid for diagonal matrices.

There are several methods for finding determinants in general. One of them is the Method of Cofactors. What follows is a very brief overview of this method. For a more detailed discussion of this method see the Method of Cofactors in the Determinants Chapter. We’ll start with a couple of definitions first.

Definition 4 If A is a square matrix then the minor of a_{i j} , denoted by M _{i j} , is the determinant of the submatrix that results from removing the i^th row and j^th column of A.

Definition 5 If A is a square matrix then the cofactor of a_{i j} , denoted by C_{i j} , is the number

( – 1)^{i +  j} M_{i j} .

Notice that the cofactor for a given entry is really just the minor for the same entry with a “+1” or a “ – 1” in front of it. The following “table” shows whether or not there should be a “+1” or a “ – 1” in front of a minor for a given cofactor.

To use the table for the cofactor C _{i j}  we simply go to the i^th row and j^th column in the table above and if there is a “+” there we leave the minor alone and if there is a “ – ” there we will tack a “ – 1” onto the appropriate minor. So, for C₃₄ we go to the 3^rd row and 4^th column and see that we have a minus sign and so we know that C₃₄ = −M₃₄ .

Here is how we can use cofactors to compute the determinant of any matrix.

Theorem 2 If A is an n × n matrix.

(a)   Choose any row, say row i, then,

det (A) = a_i1 C_i1 + a_i2 C_i2 + ··· + a_in C_in

(b)   Choose any column, say column j, then,

det (A) = a_1j  C_1j + a_2j  C_2j + ··· + a_nj C_nj.

We’ll close this review off with a significantly shortened version of Theorem 9 from Properties of Determinants section. We won’t need most of the theorem, but there are two bits of it that we’ll need so here they are. Also, there are two ways in which the theorem can be stated now that we’ve stripped out the other pieces and so we’ll give both ways of stating it here.

Theorem 3 If A is an n × n matrix then,

(a)   The only solution to the system Ax = 0 is the trivial solution (i.e. x = 0 ) if and only if det (A) ≠ 0.

(b)   The system Ax = 0 will have a non-trivial solution (i.e. x ≠ 0) if and only if det (A) = 0 .

Note that these two statements really are equivalent. Also, recall that when we say “if and only if” in a theorem statement we mean that the statement works in both directions. For example, let’s take a look at the second part of this theorem. This statement says that if Ax = 0 has nontrivial solutions then we know that we’ll also have det (A) = 0 . On the other hand, it also says that if det (A) = 0 then we’ll also know that the system will have non-trivial solutions.

This theorem will be key to allowing us to work problems in the next section.

This is then the review of determinants. Again, if you need a more detailed look at either determinants or their properties you should go back and take a look at the Determinant chapter.



Sumber

Labels: Mathematician

Thanks for reading Review of Determinants. Please share...!