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An Elementary Proof of Laplace's Formula on Determinants

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Title: An Elementary Proof of Laplace's Formula on Determinants
Language: English
Authors: Aversa, Vincenzo, De Simone, Anna
Source: International Journal of Mathematical Education in Science and Technology. 2012 43(3):418-420.
Availability: Taylor & Francis, Ltd. 325 Chestnut Street Suite 800, Philadelphia, PA 19106. Tel: 800-354-1420; Fax: 215-625-2940; Web site: http://www.tandf.co.uk/journals
Peer Reviewed: Y
Physical Description: PDF
Page Count: 3
Publication Date: 2012
Document Type: Journal Articles
Reports - Descriptive
Education Level: Higher Education
Descriptors: Geometric Concepts, College Mathematics, Undergraduate Study, Mathematics Instruction, Matrices, Validity, Mathematical Logic, Computation, Mathematical Formulas
DOI: 10.1080/0020739X.2011.592618
ISSN: 0020-739X
Abstract: A well known result due to Laplace states the equivalence between two different ways of defining the determinant of a square matrix. We give here a short proof of this result, in a form that can be presented, in our opinion, at any level of undergraduate studies.
Abstractor: As Provided
Number of References: 2
Entry Date: 2013
Accession Number: EJ992904
Database: ERIC
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  Value: <anid>AN0073443605;imt01apr.12;2019Feb27.12:35;v2.2.500</anid> <title id="AN0073443605-1">An elementary proof of Laplace's formula on determinants. </title> <p>A well known result due to Laplace states the equivalence between two different ways of defining the determinant of a square matrix. We give here a short proof of this result, in a form that can be presented, in our opinion, at any level of undergraduate studies.</p> <p>Keywords: determinant; Laplace's formula</p> <hd id="AN0073443605-2">1. Introduction</hd> <p>The attempt to simplify Laplace's theorem for the computation of the determinant of a square matrix is not fresh. A recent note is the one referred in [<reflink idref="bib1" id="ref1">1</reflink>]; others can be found on the net. Simplification is suggested by the necessity to give students of an undergraduate course in linear algebra a proof that is the least cumbersome possible. What in fact happens is that if we define determinant using permutations, then the calculus is difficult, even for a matrix of order 3; so Laplace's formula simplifies the calculus, but the teacher would like to give the proof. If we define determinant using minors, e.g. the Laplace's formula itself, we have to prove that the result of the computation does not depend on the line chosen to develop the calculus. The proof is usually omitted since it appears to be much too technical.</p> <p>In this short note we obtain a proof which is simple enough, as we hope, to be given in any elementary course. In other words, we give an alternative proof of the Laplace's expansion theorem showing that the definition of determinant based on the permutations and the one based on minors lead to the same number. This obviously suffices to prove the required independence of the latter definition from the selected line.</p> <hd id="AN0073443605-3">2. Proof of Laplace's formula</hd> <p>Let <bold>A</bold> be a square matrix with <emph>n</emph> rows and <emph>n</emph> columns. The elements of the matrix can be real numbers, complex numbers or elements belonging to any field. To fix ideas, the reader can assume them to be real numbers. Consider a product of <emph>n</emph> elements in <bold>A</bold> chosen in different rows and columns. There are <emph>n</emph> factorial ways to perform these products, one for each permutation of the set {1, 2, ... , <emph>n</emph>}; then one of them can be written as</p> <p>Graph</p> <p>where the first indices are arranged increasingly and <emph>j</emph> is a permutation of such indices. So, for any <emph>i</emph>, <emph>j</emph><subs><emph>i</emph></subs> is the column containing the sole element of the product in the row <emph>i</emph>. Two values <emph>j</emph><subs><emph>h</emph></subs> and <emph>j</emph><subs><emph>k</emph></subs> are said to form <emph>inversion</emph> in the permutation <emph>j</emph> if <emph>h</emph> < <emph>k</emph> and <emph>j</emph><subs><emph>h</emph></subs> > <emph>j</emph><subs><emph>k</emph></subs>. We denote by <emph>v</emph>(<emph>j</emph>) the number of the inversions of <emph>j</emph>.</p> <p>Denote by <emph>S</emph>(<bold>A</bold>) the sum</p> <p>Graph</p> <p>Let <bold>A</bold><subs><emph>ik</emph></subs> be the complementary minor of <emph>a</emph><subs><emph>ik</emph></subs>, that is the (<emph>n</emph> − 1) × (<emph>n</emph> − 1) matrix obtained from <bold>A</bold> by dropping the <emph>i</emph>-th row and the <emph>k</emph>-th column. The following proposition is known as 'Laplace's expansion theorem'.</p> <hd id="AN0073443605-4">Proposition</hd> <p> <emph>With the notation stated above, for every i</emph> = 1, 2, ... , <emph>n</emph>,</p> <p>Graph</p> <hd id="AN0073443605-5">Proof</hd> <p>Let <emph>b</emph><subs><emph>h</emph>ℓ</subs> be an element of <bold>A</bold><subs>ik</subs>. Obviously, an element <emph>b</emph><subs><emph>h</emph>ℓ</subs> in <bold>A</bold><subs>ik</subs> belongs to <bold>A</bold>, but it has different indices from <emph>a</emph><subs><emph>h</emph>ℓ</subs> (if <emph>h</emph> ≥ <emph>i</emph> or ℓ ≥ <emph>k</emph>, the corresponding index is increased by 1). So we write</p> <p>Graph</p> <p>where <emph>r</emph> is a permutation of {1, 2, ... , <emph>n</emph> − 1}. Our task is to prove that</p> <p>Graph</p> <p>It is not difficult to realize that, apart from the sign, each summand in the formula (<reflink idref="bib4" id="ref2">4</reflink>) is the product of <emph>n</emph> elements of the matrix <bold>A</bold> belonging to different rows and columns of <bold>A</bold>. It only remains to establish its sign. Rewriting each element with the correct indices when viewed as element of <bold>A</bold>, it is</p> <p>Graph</p> <p>where . This means that it only remains to show the equality (− 1)<sups><emph>v</emph>(<emph>s</emph>)</sups> = (−1)<sups><emph>i</emph>+<emph>k</emph>+<emph>v</emph>(<emph>r</emph>)</sups>. To this aim, comparing (<reflink idref="bib4" id="ref3">4</reflink>) and (<reflink idref="bib5" id="ref4">5</reflink>), we observe that neither the use of new indices, nor the insertion of <emph>s</emph><subs><emph>i</emph></subs> at the place <emph>i</emph> changes the number of inversions that the elements of <emph>s</emph> different from <emph>s</emph><subs><emph>i</emph></subs> form among them, this number remaining <emph>v</emph>(<emph>r</emph>); so we have to compute the number of inversions that the element <emph>s</emph><subs><emph>i</emph></subs> at place <emph>i</emph> forms with the others. Such number comes from: the number of values of <emph>s</emph> which are on the left of <emph>s</emph><subs><emph>i</emph></subs> and those numbers that are greater than <emph>i</emph>, call it <emph>b</emph>, plus the number of values of <emph>s</emph> on the right of <emph>s</emph><subs><emph>i</emph></subs> coming from numbers less than <emph>i</emph>, call it <emph>c</emph>. In symbols,</p> <p>Graph</p> <p>The requested number is then <emph>b</emph> + <emph>c</emph>. Denoting by a the number of values of <emph>s</emph> on the left of <emph>s</emph><subs><emph>i</emph></subs> coming from numbers less than <emph>i</emph>,</p> <p>Graph</p> <p>we get</p> <p>Graph</p> <p>From these equalities it follows that <emph>b</emph> + <emph>c</emph> = <emph>i</emph> + <emph>s</emph><subs><emph>i</emph></subs> − 2(<emph>a</emph> + 1). At this point, recalling that <emph>s</emph><subs><emph>i</emph></subs> equals <emph>k</emph>, the proof is complete because .□</p> <hd id="AN0073443605-6">3. Remarks</hd> <p></p> <ulist> <item> <bold> 1. In the literature the number _I_S_i_(_B_A</bold>) associated to the matrix <bold>A</bold> is known as the <emph>determinant</emph> of the matrix <bold>A</bold> and is usually denoted by |<bold>A</bold>|. Formula (<reflink idref="bib3" id="ref5">3</reflink>) furnishes a method to obtain the determinant of a matrix of order <emph>n</emph> by calculating determinants of matrices of order <emph>n</emph> − 1. As noted in the statement of the proposition, the result of the calculus does not depend on the line chosen to develop it. This implies that the right member of (<reflink idref="bib3" id="ref6">3</reflink>) can be used as an inductively definition of determinant.</item> <p></p> <item> <bold> 2. An axiomatic definition of determinant can be found in [2]. By Theorem 6.6.2 in [2] in fact, we have that a numerical function _I_f_i_(_B_A</bold>) coincides with S(<bold>A</bold>) if the following conditions hold:</item> <p></p> <item> <bold> i. the value of _I_f_i_(_B_A</bold>) remains unchanged if in the matrix <bold>A</bold> a row <emph>i</emph> is substituted by the sum of <emph>i</emph> with another row;</item> <p></p> <item> <bold> ii. if a row of _B_A</bold> is multiplied by a constant λ, the value of <emph>f</emph>(<bold>A</bold>) is multiplied by λ;</item> <p></p> <item> <bold> iii. _I_f_i_(_B_I</bold>) = 1 for the unitary matrix <bold>I</bold>.</item> </ulist> <hd id="AN0073443605-7">Acknowledgements</hd> <p>We thank an anonymous referee for the useful comments.</p> <ref id="AN0073443605-8"> <title> References </title> <blist> <bibl id="bib1" idref="ref1" type="bt">1</bibl> <bibtext> Janjiä, M. 2005. A note on Laplace's expansion theorem. Int. J. Math. Educ. Sci. Technol., 36(6): 696–698.</bibtext> </blist> <blist> <bibl id="bib2" type="bt">2</bibl> <bibtext> Mirsky, L. 1972. An Introduction to Linear Algebra, Oxford: Clarendon Press.</bibtext> </blist> </ref> <aug> <p>By Vincenzo Aversa and Anna De Simone</p> <p>Reported by Author; Author</p> </aug> <nolink nlid="nl1" bibid="bib4" firstref="ref2"></nolink> <nolink nlid="nl2" bibid="bib5" firstref="ref4"></nolink> <nolink nlid="nl3" bibid="bib3" firstref="ref5"></nolink>
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