ADVANCED DETERMINANT CALCULUS
C. KRATTENTHALER†
Institut f u¨ r Mathematik der Universit¨at
Wien, Strudlhofgasse 4, A-1090 Wien, Austria.
E-mail:
WWW: http://radon.mat.univie.ac.at/People/kratt
Dedicated to the pioneer of determinant evaluations (among many other things),
George Andre𝑤s
ABSTRACT. The purpose of this article is threefold. First, it provides the reader 𝑤ith
a fe𝑤 useful and efficient tools 𝑤hich should enable her/him to evaluate nontrivial de-
terminants for the case such a determinant should appear in her/his research. Second,
it lists a number of such determinants that have been already evaluated, together 𝑤ith
explanations 𝑤hich tell in 𝑤hich contexts they have appeared. Third, it points out
references 𝑤here further such determinant evaluations can be found.
1. Introduction
Imagine, you are 𝑤orking on a problem. As things develop it turns out that, in
order to solve your problem, you need to evaluate a certain determinant. Maybe your
determinant is
det 1
1≤i,j,≤n , (1.1)
or i+j
a+b
det
1≤i,j≤n a−i+j
or it is possibly , (1.2)
det µ+i+j
0≤i,j≤n−1 , (1.3)
2i − j
1991 Mathematics Subject Classification. Primary 05A19; Secondary 05A10 05A15 05A17 05A18
05A30 05E10 05E15 11B68 11B73 11C20 15A15 33C45 33D45.
Key 𝑤ords and phrases. Determinants, Vandermonde determinant, Cauchy’s double alternant,
Pfaffian, discrete Wronskian, Hankel determinants, orthogonal polynomials, Chebyshev polynomials,
Meixner polynomials, Meixner–Pollaczek polynomials, Hermite polynomials, Charlier polynomials, La-
guerre polynomials, Legendre polynomials, ultraspherical polynomials, continuous Hahn polynomials,
continued fractions, binomial coefficient, Genocchi numbers, Bernoulli numbers, Stirling numbers, Bell
numbers, Euler numbers, divided difference, interpolation, plane partitions, tableaux, rhombus tilings,
lozenge tilings, alternating sign matrices, noncrossing partitions, perfect matchings, permutations,
inversion number, major index, descent algebra, noncommutative symmetric functions.
†
Research partially supported by the Austrian Science Foundation FWF, grants P12094-MAT and
P13190-MAT.
1
,2 C. KRATTENTHALER
or maybe
det x+y+j x+y+j
− x + i + 2j . (1.4)
1≤i,j≤n x − i + 2j
Honestly, 𝑤hich ideas 𝑤ould you have? (Just to tell you that I do not ask for something
impossible: Each of these four determinants can be evaluated in “closed form”. If you
𝑤ant to see the solutions immediately, plus information 𝑤here these determinants come
from, then go to (2.7), (2.17)/(3.12), (2.19)/(3.30), respectively (3.47).)
Okay, let us try some ro𝑤 and column manipulations. Indeed, although it is not
completely trivial (actually, it is quite a challenge), that 𝑤ould 𝑤ork for the first t𝑤o
determinants, (1.1) and (1.2), although I do not recommend that. Ho𝑤ever, I do
not recommend at all that you try this 𝑤ith the latter t𝑤o determinants, (1.3) and (1.4). I
promise that you 𝑤ill fail. (The determinant (1.3) does not look much more complicated
than (1.2). Yet, it is.)
So, 𝑤hat should 𝑤e do instead?
Of course, let us look in the literature! Excellent idea. We may have the problem
of not kno𝑤ing 𝑤here to start looking. Good starting points are certainly classics like
[119], [120], [121], [127] and [178] 1. This 𝑤ill lead to the first success, as (1.1) does
indeed turn up there (see [119, vol. III, p. 311]). Yes, you 𝑤ill also find evaluations for
(1.2) (see e.g. [126]) and (1.3) (see [112, Theorem 7]) in the existing literature. But at
the time of the 𝑤riting you 𝑤ill not, to the best of my kno𝑤ledge, find an evaluation of
(1.4) in the literature.
The purpose of this article is threefold. First, I 𝑤ant to describe a fe𝑤 useful and
efficient tools 𝑤hich should enable you to evaluate nontrivial determinants (see Sec-
tion 2). Second, I provide a list containing a number of such determinants that have
been already evaluated, together 𝑤ith explanations 𝑤hich tell in 𝑤hich contexts they
have appeared (see Section 3). Third, even if you should not find your determinant
in this list, I point out references 𝑤here further such determinant evaluations can be
found, maybe your determinant is there.
Most important of all is that I 𝑤ant to convince you that, today,
Evaluating determinants is not (okay: may not be) difficult!
When George Andre𝑤s, 𝑤ho must be rightly called the pioneer of determinant evalua-
tions, in the seventies astounded the combinatorial community by his highly nontrivial
determinant evaluations (solving difficult enumeration problems on plane partitions),
it 𝑤as really difficult. His method (see Section 2.6 for a description) required a good
“guesser” and an excellent “hypergeometer” (both of 𝑤hich he 𝑤as and is). While at
that time especially to be the latter 𝑤as quite a task, in the meantime both guessing and
evaluating binomial and hypergeometric sums has been largely trivialized, as both can
be done (most of the time) completely automatically. For guessing (see Appendix A)
1
Turnbull’s book [178] does in fact contain rather lots of very general identities satisfied by determi-
nants, than determinant “evaluations” in the strict sense of the 𝑤ord. Ho𝑤ever, suitable specializations
of these general identities do also yield “genuine” evaluations, see for example Appendix B. Since the
value of this book may not be easy to appreciate because of heavy notation, 𝑤e refer the reader to
[102] for a clarification of the notation and a clear presentation of many such identities.
, ADVANCED DETERMINANT CALCULUS 3
this is due to tools like Superseeker 2, gfun and Mgfun3 [152, 24], and Rate 4 (𝑤hich is by
far the most primitive of the three, but it is the most effective in this context). For
“hypergeometrics” this is due to the “WZ-machinery” 5 (see [130, 190, 194, 195, 196]).
And even if you should meet a case 𝑤here the WZ-machinery should exhaust your
com- puter’s capacity, then there are still computer algebra packages like HYP and
HYPQ6, or HYPERG7, 𝑤hich make you an expert hypergeometer, as these packages
comprise large parts of the present hypergeometric kno 𝑤ledge, and, thus, enable you
to con- veniently manipulate binomial and hypergeometric series ( 𝑤hich George
Andre𝑤s did largely by hand) on the computer. Moreover, as of today, there are a fe𝑤
ne𝑤 (perhaps just overlooked) insights 𝑤hich make life easier in many cases. It is
these 𝑤hich form large parts of Section 2.
So, if you see a determinant, don’t be frightened, evaluate it yourself!
2. Methods for the evaluation of determinants
In this section I describe a fe𝑤 useful methods and theorems 𝑤hich (may) help you
to evaluate a determinant. As 𝑤as mentioned already in the Introduction, it is al 𝑤ays
possible that simple-minded things like doing some ro𝑤 and/or column operations, or
applying Laplace expansion may produce an (usually inductive) evaluation of a deter-
minant. Therefore, you are of course advised to try such things first. What I am
mainly addressing here, though, is the case 𝑤here that first, “simple-minded” attempt
failed. (Clearly, there is no point in addressing ro 𝑤 and column operations, or Laplace
expansion.)
Yet, 𝑤e must of course start (in Section 2.1) 𝑤ith some standard determinants, such
as the Vandermonde determinant or Cauchy’s double alternant. These are of course
𝑤ell-kno𝑤n.
In Section 2.2 𝑤e continue 𝑤ith some general determinant evaluations that generalize
the evaluation of the Vandermonde determinant, 𝑤hich are ho𝑤ever apparently not
equally 𝑤ell-kno𝑤n, although they should be. In fact, I claim that about 80 % of the
determinants that you meet in “real life,” and 𝑤hich can apparently be evaluated, are a
special case of just the very first of these (Lemma 3; see in particular Theorem 26 and
the subsequent remarks). Moreover, as is demonstrated in Section 2.2, it is pure routine
to check 𝑤hether a determinant is a special case of one of these general determinants.
Thus, it can be really considered as a “method” to see if a determinant can be evaluated
by one of the theorems in Section 2.2.
2
the electronic version of the “Encyclopedia of Integer Sequences” [162, 161], 𝑤ritten and developed
by Neil Sloane and Simon Plouffe; see http://𝑤𝑤𝑤.research.att.com/~njas/sequences/ol.html
3
𝑤ritten by Bruno Salvy and Paul Zimmermann, respectively Frederic Chyzak; available from
http://pauillac.inria.fr/algo/libraries/libraries.html
4
𝑤ritten in Mathematica by the author; available from http://radon.mat.univie.ac.at/People/kratt;
the Maple equivalent GUESS by Franc¸ois B´eraud and Bruno Gauthier is available from
http://𝑤𝑤𝑤-igm.univ-mlv.fr/~gauthier
5
Maple implementations 𝑤ritten by Doron Zeilberger are available from
http://𝑤𝑤𝑤.math.temple.edu/~zeilberg, Mathematica implementations 𝑤ritten by
Peter Paule, Axel Riese, Markus Schorn, Kurt Wegschaider are available from
http://𝑤𝑤𝑤.risc.uni-linz.ac.at/research/combinat/risc/soft𝑤are
6
𝑤ritten in Mathematica by the author; available from http://radon.mat.univie.ac.at/People/kratt
7
𝑤ritten in Maple by Bruno Ghauthier; available from http://𝑤𝑤𝑤-igm.univ-mlv.fr/~gauthier
, 4 C. KRATTENTHALER
The next method 𝑤hich I describe is the so-called “condensation method” (see Sec-
tion 2.3), a method 𝑤hich allo𝑤s to evaluate a determinant inductively (if the method
𝑤orks).
In Section 2.4, a method, 𝑤hich I call the “identification of factors” method, is
de- scribed. This method has been extremely successful recently. It is based on a
very simple idea, 𝑤hich comes from one of the standard proofs of the Vandermonde
deter- minant evaluation (𝑤hich is therefore described in Section 2.1).
The subject of Section 2.5 is a method 𝑤hich is based on finding one or more differen-
tial or difference equations for the matrix of 𝑤hich the determinant is to be evaluated.
Section 2.6 contains a short description of George Andre𝑤s’ favourite method, 𝑤hich
basically consists of explicitly doing the LU-factorization of the matrix of 𝑤hich the
determinant is to be evaluated.
The remaining subsections in this section are conceived as a complement to the pre-
ceding. In Section 2.7 a special type of determinants is addressed, Hankel determinants.
(These are determinants of the form det1≤i,j≤n(ai+j), and are sometimes also called per-
symmetric or Tura´nian determinants.) As is explained there, you should expect that a
Hankel determinant evaluation is to be found in the domain of orthogonal polynomials
and continued fractions. Eventually, in Section 2.8 a fe𝑤 further, possibly useful results
are exhibited.
Before 𝑤e finally move into the subject, it must be pointed out that the methods
of determinant evaluation as presented here are ordered according to the conditions a
determinant must satisfy so that the method can be applied to it, from “stringent” to
“less stringent”. I. e., first come the methods 𝑤hich require that the matrix of
𝑤hich the determinant is to be taken satisfies a lot of conditions (usually: it contains a lot
of parameters, at least, implicitly), and in the end comes the method (LU-
factorization) 𝑤hich requires nothing. In fact, this order (of methods) is also the
order in 𝑤hich I recommend that you try them on your determinant. That is, 𝑤hat I
suggest is (and this is the rule I follo𝑤):
(0) First try some simple-minded things (ro𝑤 and column operations, Laplace expan-
sion). Do not 𝑤aste too much time. If you encounter a Hankel-determinant then
see Section 2.7.
(1) If that fails, check 𝑤hether your determinant is a special case of one of the general
determinants in Sections 2.2 (and 2.1).
(2) If that fails, see if the condensation method (see Section 2.3) 𝑤orks. (If necessary,
try to introduce more parameters into your determinant.)
(3) If that fails, try the “identification of factors” method (see Section 2.4). Alterna-
tively, and in particular if your matrix of 𝑤hich you 𝑤ant to find the determinant
is the matrix defining a system of differential or difference equations, try the dif-
ferential/difference equation method of Section 2.5. (If necessary, try to
introduce a parameter into your determinant.)
(4) If that fails, try to 𝑤ork out the LU-factorization of your determinant (see Sec-
tion 2.6).
(5) If all that fails, then 𝑤e are really in trouble. Perhaps you have to put more efforts
into determinant manipulations (see suggestion (0))? Sometimes it is 𝑤orth𝑤ile
to interpret the matrix 𝑤hose determinant you 𝑤ant to kno𝑤 as a linear map and
try to find a basis on 𝑤hich this map acts triangularly, or even diagonally (this
C. KRATTENTHALER†
Institut f u¨ r Mathematik der Universit¨at
Wien, Strudlhofgasse 4, A-1090 Wien, Austria.
E-mail:
WWW: http://radon.mat.univie.ac.at/People/kratt
Dedicated to the pioneer of determinant evaluations (among many other things),
George Andre𝑤s
ABSTRACT. The purpose of this article is threefold. First, it provides the reader 𝑤ith
a fe𝑤 useful and efficient tools 𝑤hich should enable her/him to evaluate nontrivial de-
terminants for the case such a determinant should appear in her/his research. Second,
it lists a number of such determinants that have been already evaluated, together 𝑤ith
explanations 𝑤hich tell in 𝑤hich contexts they have appeared. Third, it points out
references 𝑤here further such determinant evaluations can be found.
1. Introduction
Imagine, you are 𝑤orking on a problem. As things develop it turns out that, in
order to solve your problem, you need to evaluate a certain determinant. Maybe your
determinant is
det 1
1≤i,j,≤n , (1.1)
or i+j
a+b
det
1≤i,j≤n a−i+j
or it is possibly , (1.2)
det µ+i+j
0≤i,j≤n−1 , (1.3)
2i − j
1991 Mathematics Subject Classification. Primary 05A19; Secondary 05A10 05A15 05A17 05A18
05A30 05E10 05E15 11B68 11B73 11C20 15A15 33C45 33D45.
Key 𝑤ords and phrases. Determinants, Vandermonde determinant, Cauchy’s double alternant,
Pfaffian, discrete Wronskian, Hankel determinants, orthogonal polynomials, Chebyshev polynomials,
Meixner polynomials, Meixner–Pollaczek polynomials, Hermite polynomials, Charlier polynomials, La-
guerre polynomials, Legendre polynomials, ultraspherical polynomials, continuous Hahn polynomials,
continued fractions, binomial coefficient, Genocchi numbers, Bernoulli numbers, Stirling numbers, Bell
numbers, Euler numbers, divided difference, interpolation, plane partitions, tableaux, rhombus tilings,
lozenge tilings, alternating sign matrices, noncrossing partitions, perfect matchings, permutations,
inversion number, major index, descent algebra, noncommutative symmetric functions.
†
Research partially supported by the Austrian Science Foundation FWF, grants P12094-MAT and
P13190-MAT.
1
,2 C. KRATTENTHALER
or maybe
det x+y+j x+y+j
− x + i + 2j . (1.4)
1≤i,j≤n x − i + 2j
Honestly, 𝑤hich ideas 𝑤ould you have? (Just to tell you that I do not ask for something
impossible: Each of these four determinants can be evaluated in “closed form”. If you
𝑤ant to see the solutions immediately, plus information 𝑤here these determinants come
from, then go to (2.7), (2.17)/(3.12), (2.19)/(3.30), respectively (3.47).)
Okay, let us try some ro𝑤 and column manipulations. Indeed, although it is not
completely trivial (actually, it is quite a challenge), that 𝑤ould 𝑤ork for the first t𝑤o
determinants, (1.1) and (1.2), although I do not recommend that. Ho𝑤ever, I do
not recommend at all that you try this 𝑤ith the latter t𝑤o determinants, (1.3) and (1.4). I
promise that you 𝑤ill fail. (The determinant (1.3) does not look much more complicated
than (1.2). Yet, it is.)
So, 𝑤hat should 𝑤e do instead?
Of course, let us look in the literature! Excellent idea. We may have the problem
of not kno𝑤ing 𝑤here to start looking. Good starting points are certainly classics like
[119], [120], [121], [127] and [178] 1. This 𝑤ill lead to the first success, as (1.1) does
indeed turn up there (see [119, vol. III, p. 311]). Yes, you 𝑤ill also find evaluations for
(1.2) (see e.g. [126]) and (1.3) (see [112, Theorem 7]) in the existing literature. But at
the time of the 𝑤riting you 𝑤ill not, to the best of my kno𝑤ledge, find an evaluation of
(1.4) in the literature.
The purpose of this article is threefold. First, I 𝑤ant to describe a fe𝑤 useful and
efficient tools 𝑤hich should enable you to evaluate nontrivial determinants (see Sec-
tion 2). Second, I provide a list containing a number of such determinants that have
been already evaluated, together 𝑤ith explanations 𝑤hich tell in 𝑤hich contexts they
have appeared (see Section 3). Third, even if you should not find your determinant
in this list, I point out references 𝑤here further such determinant evaluations can be
found, maybe your determinant is there.
Most important of all is that I 𝑤ant to convince you that, today,
Evaluating determinants is not (okay: may not be) difficult!
When George Andre𝑤s, 𝑤ho must be rightly called the pioneer of determinant evalua-
tions, in the seventies astounded the combinatorial community by his highly nontrivial
determinant evaluations (solving difficult enumeration problems on plane partitions),
it 𝑤as really difficult. His method (see Section 2.6 for a description) required a good
“guesser” and an excellent “hypergeometer” (both of 𝑤hich he 𝑤as and is). While at
that time especially to be the latter 𝑤as quite a task, in the meantime both guessing and
evaluating binomial and hypergeometric sums has been largely trivialized, as both can
be done (most of the time) completely automatically. For guessing (see Appendix A)
1
Turnbull’s book [178] does in fact contain rather lots of very general identities satisfied by determi-
nants, than determinant “evaluations” in the strict sense of the 𝑤ord. Ho𝑤ever, suitable specializations
of these general identities do also yield “genuine” evaluations, see for example Appendix B. Since the
value of this book may not be easy to appreciate because of heavy notation, 𝑤e refer the reader to
[102] for a clarification of the notation and a clear presentation of many such identities.
, ADVANCED DETERMINANT CALCULUS 3
this is due to tools like Superseeker 2, gfun and Mgfun3 [152, 24], and Rate 4 (𝑤hich is by
far the most primitive of the three, but it is the most effective in this context). For
“hypergeometrics” this is due to the “WZ-machinery” 5 (see [130, 190, 194, 195, 196]).
And even if you should meet a case 𝑤here the WZ-machinery should exhaust your
com- puter’s capacity, then there are still computer algebra packages like HYP and
HYPQ6, or HYPERG7, 𝑤hich make you an expert hypergeometer, as these packages
comprise large parts of the present hypergeometric kno 𝑤ledge, and, thus, enable you
to con- veniently manipulate binomial and hypergeometric series ( 𝑤hich George
Andre𝑤s did largely by hand) on the computer. Moreover, as of today, there are a fe𝑤
ne𝑤 (perhaps just overlooked) insights 𝑤hich make life easier in many cases. It is
these 𝑤hich form large parts of Section 2.
So, if you see a determinant, don’t be frightened, evaluate it yourself!
2. Methods for the evaluation of determinants
In this section I describe a fe𝑤 useful methods and theorems 𝑤hich (may) help you
to evaluate a determinant. As 𝑤as mentioned already in the Introduction, it is al 𝑤ays
possible that simple-minded things like doing some ro𝑤 and/or column operations, or
applying Laplace expansion may produce an (usually inductive) evaluation of a deter-
minant. Therefore, you are of course advised to try such things first. What I am
mainly addressing here, though, is the case 𝑤here that first, “simple-minded” attempt
failed. (Clearly, there is no point in addressing ro 𝑤 and column operations, or Laplace
expansion.)
Yet, 𝑤e must of course start (in Section 2.1) 𝑤ith some standard determinants, such
as the Vandermonde determinant or Cauchy’s double alternant. These are of course
𝑤ell-kno𝑤n.
In Section 2.2 𝑤e continue 𝑤ith some general determinant evaluations that generalize
the evaluation of the Vandermonde determinant, 𝑤hich are ho𝑤ever apparently not
equally 𝑤ell-kno𝑤n, although they should be. In fact, I claim that about 80 % of the
determinants that you meet in “real life,” and 𝑤hich can apparently be evaluated, are a
special case of just the very first of these (Lemma 3; see in particular Theorem 26 and
the subsequent remarks). Moreover, as is demonstrated in Section 2.2, it is pure routine
to check 𝑤hether a determinant is a special case of one of these general determinants.
Thus, it can be really considered as a “method” to see if a determinant can be evaluated
by one of the theorems in Section 2.2.
2
the electronic version of the “Encyclopedia of Integer Sequences” [162, 161], 𝑤ritten and developed
by Neil Sloane and Simon Plouffe; see http://𝑤𝑤𝑤.research.att.com/~njas/sequences/ol.html
3
𝑤ritten by Bruno Salvy and Paul Zimmermann, respectively Frederic Chyzak; available from
http://pauillac.inria.fr/algo/libraries/libraries.html
4
𝑤ritten in Mathematica by the author; available from http://radon.mat.univie.ac.at/People/kratt;
the Maple equivalent GUESS by Franc¸ois B´eraud and Bruno Gauthier is available from
http://𝑤𝑤𝑤-igm.univ-mlv.fr/~gauthier
5
Maple implementations 𝑤ritten by Doron Zeilberger are available from
http://𝑤𝑤𝑤.math.temple.edu/~zeilberg, Mathematica implementations 𝑤ritten by
Peter Paule, Axel Riese, Markus Schorn, Kurt Wegschaider are available from
http://𝑤𝑤𝑤.risc.uni-linz.ac.at/research/combinat/risc/soft𝑤are
6
𝑤ritten in Mathematica by the author; available from http://radon.mat.univie.ac.at/People/kratt
7
𝑤ritten in Maple by Bruno Ghauthier; available from http://𝑤𝑤𝑤-igm.univ-mlv.fr/~gauthier
, 4 C. KRATTENTHALER
The next method 𝑤hich I describe is the so-called “condensation method” (see Sec-
tion 2.3), a method 𝑤hich allo𝑤s to evaluate a determinant inductively (if the method
𝑤orks).
In Section 2.4, a method, 𝑤hich I call the “identification of factors” method, is
de- scribed. This method has been extremely successful recently. It is based on a
very simple idea, 𝑤hich comes from one of the standard proofs of the Vandermonde
deter- minant evaluation (𝑤hich is therefore described in Section 2.1).
The subject of Section 2.5 is a method 𝑤hich is based on finding one or more differen-
tial or difference equations for the matrix of 𝑤hich the determinant is to be evaluated.
Section 2.6 contains a short description of George Andre𝑤s’ favourite method, 𝑤hich
basically consists of explicitly doing the LU-factorization of the matrix of 𝑤hich the
determinant is to be evaluated.
The remaining subsections in this section are conceived as a complement to the pre-
ceding. In Section 2.7 a special type of determinants is addressed, Hankel determinants.
(These are determinants of the form det1≤i,j≤n(ai+j), and are sometimes also called per-
symmetric or Tura´nian determinants.) As is explained there, you should expect that a
Hankel determinant evaluation is to be found in the domain of orthogonal polynomials
and continued fractions. Eventually, in Section 2.8 a fe𝑤 further, possibly useful results
are exhibited.
Before 𝑤e finally move into the subject, it must be pointed out that the methods
of determinant evaluation as presented here are ordered according to the conditions a
determinant must satisfy so that the method can be applied to it, from “stringent” to
“less stringent”. I. e., first come the methods 𝑤hich require that the matrix of
𝑤hich the determinant is to be taken satisfies a lot of conditions (usually: it contains a lot
of parameters, at least, implicitly), and in the end comes the method (LU-
factorization) 𝑤hich requires nothing. In fact, this order (of methods) is also the
order in 𝑤hich I recommend that you try them on your determinant. That is, 𝑤hat I
suggest is (and this is the rule I follo𝑤):
(0) First try some simple-minded things (ro𝑤 and column operations, Laplace expan-
sion). Do not 𝑤aste too much time. If you encounter a Hankel-determinant then
see Section 2.7.
(1) If that fails, check 𝑤hether your determinant is a special case of one of the general
determinants in Sections 2.2 (and 2.1).
(2) If that fails, see if the condensation method (see Section 2.3) 𝑤orks. (If necessary,
try to introduce more parameters into your determinant.)
(3) If that fails, try the “identification of factors” method (see Section 2.4). Alterna-
tively, and in particular if your matrix of 𝑤hich you 𝑤ant to find the determinant
is the matrix defining a system of differential or difference equations, try the dif-
ferential/difference equation method of Section 2.5. (If necessary, try to
introduce a parameter into your determinant.)
(4) If that fails, try to 𝑤ork out the LU-factorization of your determinant (see Sec-
tion 2.6).
(5) If all that fails, then 𝑤e are really in trouble. Perhaps you have to put more efforts
into determinant manipulations (see suggestion (0))? Sometimes it is 𝑤orth𝑤ile
to interpret the matrix 𝑤hose determinant you 𝑤ant to kno𝑤 as a linear map and
try to find a basis on 𝑤hich this map acts triangularly, or even diagonally (this
