Chapter 4 Determinants 4.1 Definition Using Expansion by Minors Every square matrix A has a number associated to it and called its determinant,denotedbydet(A). One of the most important properties of a determinant is that it gives us a criterion to decide whether the matrix is invertible: A matrix A is invertible i↵ det(A) =0. 6 It is possible to define determinants in terms of a fairly complicated formula involving n!terms(assumingA is a n n matrix) but this way to proceed makes it more difficult⇥ to prove properties of determinants. 331 332 CHAPTER 4. DETERMINANTS Consequently, we follow a more algorithmic approach due to Mike Artin. We will view the determinant as a function of the rows of an n n matrix. ⇥ Formally, this means that n n det: (R ) R. ! We will define the determinant recursively using a pro- cess called expansion by minors. Then, we will derive properties of the determinant and prove that there is a unique function satisfying these prop- erties. As a consequence, we will have an axiomatic definition of the determinant. 4.1. DEFINITION USING EXPANSION BY MINORS 333 For a 1 1matrixA =(a), we have ⇥ det(A)=det(a)=a. For a 2 2matrix, ⇥ ab A = cd it will turn out that det(A)=ad bc. − The determinant has a geometric interpretation as a signed area, in higher dimension as a signed volume. In order to describe the recursive process to define a de- terminant we need the notion of a minor. 334 CHAPTER 4. DETERMINANTS Definition 4.1. Given any n n matrix with n 2, ⇥ ≥ for any two indices i, j with 1 i, j n,letAij be the (n 1) (n 1) matrix obtained by deleting row i and colummn− ⇥j from− A and called a minor: ⇥ 2 ⇥ 3 ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ ⇥ A = 6 7 ij 6 ⇥ 7 6 7 6 ⇥ 7 6 7 6 ⇥ 7 6 7 6 ⇥ 7 For example, if 4 5 2 1 0 00 1 −2 1 0 0 2− − 3 A = 0 1 2 10 − − 6 00 1 2 17 6 − − 7 6 000 127 6 − 7 then 4 5 2 10 0 0 −1 10 A = . 23 200− − 2 13 − 600 127 6 − 7 4 5 4.1. DEFINITION USING EXPANSION BY MINORS 335 We can now proceed with the definition of determinants. Definition 4.2. Given any n n matrix A =(aij), if n =1,then ⇥ det(A)=a11, else det(A)=a det(A )+ +( 1)i+1a det(A )+ 11 11 ··· − i1 i1 ··· +( 1)n+1a det(A ), ( ) − n1 n1 ⇤ the expansion by minors on the first column. When n =2,wehave a11 a12 det = a11 det[a22] a21 det[a12]=a11a22 a21a12, a21 a22 − − which confirms the formula claimed earlier. 336 CHAPTER 4. DETERMINANTS When n =3,weget a11 a12 a13 a22 a23 a12 a13 det a21 a22 a23 = a11 det a21 det 2 3 a32 a33 − a32 a33 a31 a32 a33 4 5 a a + a det 12 13 , 31 a a 22 23 and using the formula for a 2 2determinant,weget ⇥ a11 a12 a13 det a a a 2 21 22 233 a31 a32 a33 4= a (a a 5 a a ) a (a a a a ) 11 22 33 − 32 23 − 21 12 33 − 32 13 + a (a a a a ). 31 12 23 − 22 13 As we can see, the formula is already quite complicated! 4.1. DEFINITION USING EXPANSION BY MINORS 337 Given a n n-matrix A =(aij), its determinant det(A) is also denoted⇥ by a11 a12 ... a1 n a a ... a det(A)= 21 22 2 n . . ... a a ... a n 1 n 2 nn We now derive some important and useful properties of the determinant. Recall that we view the determinant det(A)asafunction of the rows of the matrix A,sowecanwrite det(A)=det(A1,...,An), where A1,...,An are the rows of A. 338 CHAPTER 4. DETERMINANTS Proposition 4.1. The determinant function det: (Rn)n R satisfies the following properties: ! (1) det(I)=1, where I is the identity matrix. (2) The determinant is linear in each of its rows; this means that det(A1,...,Ai 1,B+ C, Ai+1,...,An)= − det(A1,...,Ai 1,B,Ai+1,...,An) − +det(A1,...,Ai 1,C,Ai+1,...,An) − and det(A1,...,Ai 1,λAi,Ai+1,...,An)= − λ det(A1,...,Ai 1,Ai,Ai+1,...,An). − (3) If two adjacent rows of A are equal, then det(A)=0. This means that det(A1,...,Ai,Ai,...,An)=0. Property (2) says that det is a multilinear map,and property (3) says that det is an alternating map. 4.1. DEFINITION USING EXPANSION BY MINORS 339 Proposition 4.2. The determinant function det: (Rn)n R satisfies the following properties: ! (4) If two adjacent rows are interchanged, then the de- terminant is multiplied by 1; thus, − det(A1,...,Ai+1,Ai,...,An) = det(A ,...,A,A ,...,A ). − 1 i i+1 n (5) If two rows are identical then the determinant is zero; that is, det(A1,...,Ai,...,Ai,...,An)=0. (6) If any two distinct rows of A are are interchanged, then the determinant is multiplied by 1; thus, − det(A1,...,Aj,...,Ai,...,An) = det(A ,...,A,...,A ,...,A ). − 1 i j n (7) If a multiple of a row is added to another row, the determinant is unchanged; that is, det(A1,...,Ai + λAj,...,An) =det(A1,...,Ai,...,An). (8) If any row of A is zero, then det(A)=0. 340 CHAPTER 4. DETERMINANTS Using property (6), it is easy to show that the expansion by minors formula ( )canbeadpatedtoanycolumn. Indeed, we have ⇤ det(A)=( 1)j+1a det(A )+ +( 1)j+ia det(A ) − 1j 1j ··· − ij ij + +( 1)j+na det(A ). ( ) ··· − nj nj ⇤⇤ The beauty of this approach is that properties (6) and (7) describe the e↵ect of the elementary operations P (i, j) and Ei,j,λ on the determinant: Indeed, (6) says that det(P (i, j)A)= det(A), (a) − and (7) says that det(Ei,j;λA)=det(A). (b) Furthermore, linearity (propery (2)) says that det(Ei,λA)=λ det(A). (c) 4.1. DEFINITION USING EXPANSION BY MINORS 341 Substituting the identity I for A in the above equations, since det(I)=1,wefindthedeterminantsoftheelemen- tary matrices: (1) For any permutation matrix P (i, j)(i = j), we have 6 det(P (i, j)) = 1. − (2) For any row operation Ei,j;λ (adding λ times row j to row i), we have det(Ei,j;λ)=1. (3) For any row operation Ei,λ (multiplying row i by λ), we have det(Ei,λ)=λ. The above properties together with the equations (a), (b), (c) yield the following important proposition: 342 CHAPTER 4. DETERMINANTS Proposition 4.3. For every n n matrix A and every elementary matrix E, we have⇥ det(EA)=det(E)det(A). We can now use Proposition 4.3 and the reduction to row echelon form to compute det(A). Indeed, recall that we showed (just before Proposition 2.15)) that every square matrix A can be reduced by el- ementary operations to a matrix A0 which is either the identity or else whose last row is zero, A0 = E E A. k ··· 1 If A0 = I,thendet(A0)=1by(1),elseisA0 has a zero row, then det(A0)=0by(8). 4.1. DEFINITION USING EXPANSION BY MINORS 343 Furthermore, by induction using Proposition 4.3 (see the proof of Proposition 4.7), we get det(A0)=det(E E A)=det(E ) det(E )det(A). k ··· 1 k ··· 1 Since all the determinants, det(Ek)oftheelementaryma- trices Ei are known, we see that the formula det(A0)=det(E ) det(E )det(A) k ··· 1 determines A.Asaconsequence,wehavethefollowing characterization of a determinant: Theorem 4.4. (Axiomatic Characterization of the De- terminant) The determinant det is the unique func- tion f :(Rn)n R satisfying properties (1), (2), and (3) of Proposition! 4.1. 344 CHAPTER 4. DETERMINANTS Instead of evaluating a determinant using expansion by minors on the columns, we can use expansion by minors on the rows. Indeed, define the function D given D(A)=( 1)i+1a D(A )+ − i 1 i 1 ··· +( 1)i+na D(A ), ( ) − in in † with D([a]) = a. Then, it is fairly easy to show that the properties of Proposition 4.1 hold for D,andthus,byTheorem4.4, the function D also defines the determinant, that is, D(A)=det(A). Proposition 4.5. For any square matrix A, we have det(A)=det(A>). 4.1. DEFINITION USING EXPANSION BY MINORS 345 We also obtain the important characterization of invert- ibility of a matrix in terms of its determinant. Proposition 4.6. A square matrix A is invertible i↵ det(A) =0. 6 We can now prove one of the most useful properties of determinants. Proposition 4.7. Given any two n n matrices A and B, we have ⇥ det(AB)=det(A)det(B). In order to give an explicit formula for the determinant, we need to discuss some properties of permutation matri- ces. 346 CHAPTER 4. DETERMINANTS 4.2 Permutations and Permutation Matrices Let [n]= 1, 2 ...,n ,wheren N,andn>0. { } 2 Definition 4.3. A permutation on n elements is a bi- jection ⇡ :[n] [n]. When n =1,theonlyfunction from [1] to [1]! is the constant map: 1 1. Thus, we will assume that n 2. A transposition7! is a permu- tation ⌧ :[n] [n]suchthat,forsome≥ i<j(with 1 i<j !n), ⌧(i)=j, ⌧(j)=i,and⌧(k)=k, for all k [n] i, j .Inotherwords,atransposition exchanges2 two distinct−{ } elements i, j [n].