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As described above, we want a formulato determine whether an matrix is nonsingular.We will not begin by stating such a formula.Instead, we will begin by considering the function that such a formulacalculates.We will define the function by its properties,then prove that the function with these properties exists and is uniqueand also describe formulas that compute this function.(Because we will show that the function exists and is unique, from the start we will say "" instead of"if there is a determinant function then " and"the determinant" instead of "any determinant".)
- Definition 2.1
A determinant is a function such that
- for
- for
- for
- where is an identity matrix
(the 's are the rows of the matrix). We often write for .
- Remark 2.2
Property (2) is redundant since
swaps rows and . It is listed only for convenience.
The first result shows that a function satisfying these conditionsgives a criteria for nonsingularity.(Its last sentence is that, in the context of the firstthree conditions, (4) is equivalent to the conditionthat the determinant of an echelon form matrix is the product down the diagonal.)
- Lemma 2.3
A matrix with two identical rows has a determinant of zero.A matrix with a zero row has a determinant of zero.A matrix is nonsingular if and only if its determinant is nonzero.The determinant of an echelon form matrix is the product down its diagonal.
- Proof
To verify the first sentence, swap the two equal rows.The sign of the determinant changes, but the matrix is unchangedand so its determinant is unchanged.Thus the determinant is zero.
For the second sentence, we multiply a zero row by −1 and apply property (3). Multiplying a zero row with a constant leaves the matrix unchanged, so property (3) implies that . The only way this can be is if .
For the third sentence, where is theGauss-Jordan reduction, by the definitionthe determinant of is zero if and only ifthe determinant of is zero(although they could differ in sign or magnitude).A nonsingular Gauss-Jordan reduces to an identity matrixand so has a nonzero determinant.A singular reduces to a with a zero row;by the second sentence of this lemma its determinant is zero.
Finally,for the fourth sentence, if an echelon form matrix is singular thenit has a zero on its diagonal, that is,the product down its diagonal is zero.The third sentence says that if a matrix is singular then its determinant is zero.So if the echelon form matrix is singular then its determinant equals theproduct down its diagonal.
If an echelon form matrix is nonsingular then none of its diagonal entriesis zero so we can use property (3) of the definition to factor them out(again, the vertical bars indicate the determinantoperation).
Next, the Jordan half of Gauss-Jordan elimination,using property (1) of the definition, leaves the identity matrix.
Therefore, if an echelon form matrix is nonsingular then its determinant is the product down its diagonal.
That result gives us a way to compute the value of a determinantfunction on a matrix.Do Gaussian reduction, keeping track of any changes of sign caused by row swaps and any scalars that are factored out, and then finish by multiplyingdown the diagonal of the echelon form result.This procedure takes the same time as Gauss' method and sois sufficiently fast to be practical on the size matrices that we see in this book.
- Example 2.4
Doing determinants
with Gauss' method won't give a big savingsbecause the determinant formula is so easy.However, a determinant is usually easier to calculatewith Gauss' method than with the formula given earlier.
- Example 2.5
Determinants of matrices any bigger than are almost always most quickly done with this Gauss' method procedure.
The prior example illustrates an important point.Although we have not yet found a determinant formula,if one exists then we know what value it gives to the matrix — if there is a function with properties (1)-(4) then on the above matrix the function must return .
- Lemma 2.6
For each , if there is an determinant function then it is unique.
- Proof
For any matrix we can perform Gauss' method on the matrix, keeping track of how the sign alternates on row swaps, and then multiply down the diagonal of the echelon form result. By the definition and the lemma, all determinant functions must return this value on this matrix. Thus all determinant functions are equal, that is, there is only one input argument/output value relationship satisfying the four conditions.
The "if there is an determinant function" emphasizes that,although we canuse Gauss' method to compute the only value that a determinant function could possibly return, we haven't yet shown that such a determinant function exists for all .In the rest of the section we will produce determinant functions.
Exercises[edit | edit source]
For these, assume that an determinant function exists for all .
- This exercise is recommended for all readers.
- Problem 1
Use Gauss' method to find each determinant.
- Problem 2
- Use Gauss' method to find each.
- Problem 3
For which values of does this system have a unique solution?
- This exercise is recommended for all readers.
- Problem 4
Express each of these in terms of .
- This exercise is recommended for all readers.
- Problem 5
Find the determinant of a diagonal matrix.
- Problem 6
Describe the solution set of a homogeneous linear system if thedeterminant of the matrix of coefficients is nonzero.
- This exercise is recommended for all readers.
- Problem 7
Show that this determinant is zero.
- Problem 8
- Find the , , and matriceswith entry given by .
- Find the determinant of the square matrix with entry .
- Problem 9
- Find the , , and matriceswith entry given by .
- Find the determinant of the square matrix with entry .
- This exercise is recommended for all readers.
- Problem 10
Show that determinant functions are not linear by giving a case where .
- Problem 11
The second condition in the definition, that row swaps change the sign of a determinant, is somewhat annoying.It means we have to keep track of the number of swaps, to compute howthe sign alternates.Can we get rid of it?Can we replace it with the condition that row swaps leave the determinantunchanged?(If so then we would need new ,, and formulas, but that would be a minor matter.)
- Problem 12
Prove that the determinant of any triangular matrix, upper or lower,is the product down its diagonal.
- Problem 13
Refer to the definition of elementary matrices in the Mechanicsof Matrix Multiplication subsection.
- What is the determinant of each kind of elementary matrix?
- Prove that if is any elementary matrix then for any appropriately sized.
- (This question doesn't involve determinants.)Prove that if is singular then a product isalso singular.
- Show that .
- Show that if is nonsingular then.
- Problem 14
Prove that the determinant of a product is the product of thedeterminants in this way.Fix the matrix and consider the function given by.
- Check that satisfies property (1) in the definition ofa determinant function.
- Check property (2).
- Check property (3).
- Check property (4).
- Conclude the determinant of a product is the product of thedeterminants.
- Problem 15
A submatrix of a given matrix is one that can beobtained by deleting some of the rows and columns of. Thus, the first matrix here is a submatrix of thesecond.
Prove that for any square matrix,the rank of the matrix is if and only if is the largestinteger such that there is an submatrix with a nonzerodeterminant.
- This exercise is recommended for all readers.
- Problem 16
Prove that a matrix with rational entries has a rational determinant.
- ? Problem 17
Find the element of likeness in (a) simplifying a fraction, (b) powderingthe nose, (c) building new steps on the church, (d) keeping emeritusprofessors on campus, (e) putting , , in thedeterminant
()
Solutions
References[edit | edit source]
- Anning, Norman (proposer); Trigg, C. W. (solver) (1953), "Elementary problem 1016", American Mathematical Monthly, American Mathematical Society, 60 (2): 115
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ignored (help).
Linear Algebra | ||
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