Math 1b Practical — Determinan tsJanuary 30, 2011Square matrices have determinants, which are scalars. Determinants can be intro-duced in several ways; we choose to give a recursive definition. The determinant of a 1 × 1matrix is the entry of the matrix. Once we have defined the determinant of (n−1)×(n−1)matrices, we define the determinant of an n × n matrix A with entries aijasdet(A)=a11det(A11) − a12det(A12)+...+(−1)n−1a1ndet(A1n).Here Aijdenotes the submatrix of A obtained by deleting row i and column j from A.It can be seen inductively that the terms (monomials) that appear in det(A)areproducts a1,j1a2,j2···an,jnwhere j1,j2,...,jnare 1, 2,...,n in some order, each witha coefficient of +1 or −1. Such a sequence j1,j2,...,jnmay be called a permutation of{1, 2,...,n}, and the coefficient of the term a1,j1a2,j2···an,jnin the determinant expansionof A is called the sign of the permutation.For example, when n =4,theterma13a21a34a42arises asdet⎛⎜⎝a11a12a13a14a21a22a23a24a31a32a33a34a41a42a34a44⎞⎟⎠= ...+(+1)a13det⎛⎝a21a22a24a31a32a34a41a42a44⎞⎠+ ...= ...+ (+1)(+1)a13a21deta32a34a42a44+ ...= ...+ (+1)(+1)(−1)a13a21a34det ( a42)+...= ...+ (+1)(+1)(−1)(+1)a13a21a34a42+ ....So the sign of 3, 1, 4, 2is−1.The following rule for computing the sign of a permutation may be extracted from themethod illustrated above. [There are other approaches to understanding signs, and youmay use any of them.] Given a permutation j1,j2,...,jn, write a sign +1 under jiwhen jiis in an odd-numbered position when ji,ji+1,...jnare rearranged in increasing numericalorder, and a sign −1whenjiis in an even-numbered position. Then the sign of thepermutation is the product of the signs under the ji’s. For example, when j1,j2,...,jn=2, 6, 4, 1, 5, 3, we get264153− +++− +,so the sign of the permutation is +. The sign under the 4, for example, is + because 4 isin the third (an odd) position when 4, 1, 5, 3 is reordered as 1, 3, 4, 5.The following rules are extremely important. Some explanation of why they hold willbe given later, but for the moment we just apply them.(o) The determinant of the identity matrix is 1.(i) If the matrix Ais obtained from A by interchanging two rows of A,thendet(A)=− det(A).(ii) If the matrix Ais obtained from A by multiplying a row of A by a scalar t,thendet(A)=t det(A).(iii) If the matrix Ais obtained from A by adding a scalar multiple of one row of Ato another, then det(A)=det(A).These rules allow the computation of the determinant of a matrix A by reducing it toechelon form while keeping track of how the determinant changes with each row operation.The proofs below are just sketches. We will fill in details and do examples in class.Theorem 1. A square matrix A is nonsingular if and only if det(A) =0.Proof: Since A is square, the echelon form Aof A is either I of determinant 1 (when Ais nonsingular), or has a row of all zeros at the bottom and so of determinant 0 (whenA is singular). By the Rules, the determinant of A will be a nonzero scalar times thedeterminant of A.SoitisnonzeroifA is nonsingular, and zero if A is singular. Theorem 2. The determinant of an upper triangular matrix T is the product of itsdiagonal elements.Proof: From Rule (ii), the determinant of a diagonal matrix is the product of the diagonalentries [examples in class].If an upper triangular matrix T has a zero on its diagonal, then its echelon form willhave a row of all zeros and Rule (ii) shows that its determinant is 0. If all diagonal entriesof T are nonzero, then Using only row operations of Type (iii), which do not change thedeterminant, the matrix can be made diagonal without changing the diagonal entries. Theorem 3. For square matrices A and B of the same size, det(AB)=det(A)det(B).Proof: The proof will be sketched in class. First one checks that the statement is truewhen A is an elementary matrix. The we use the fact that every nonsingular matrix A isthe product of elementary matrices. When A is singular, it can be seen that both det(A)and det(AB)are0. Theorem 4. For any square matrix A, det(A )=det(A).Proof: First check that det(E )=det(E) for elementary matrices E.IfA is singular, bothdet(A) and det(A )are0. IfA is nonsingular, then it is the product E1E2···Erof somenumber of elementary matrices. Using the theorems above, and the rule (BC) = C B ,det(A )=det(E rE r−1···E 1)=det(E r)det(E r−1) ···det(E 1)=det(Er)det(Er−1) ···det(E1)=det(E1)det(E2) ···det(Er)=det(E1E2···Er)=det(A). Corollary. The determinant of a lower triangular matrix T is the product of its diagonalelements.Theorem 5 . The k-dimensional volume V of the parallelopiped spanned by the rows ofa k by n matrix M is the square- ro ot of det(MM ).Proof: If the rows of M are linearly dependent, both det(MM )=0andV =0,soassumeM has rank k.Let v1, v2,...,vkbe the rows of M.Letu1, u2,...,ukbe the vectors obtained bythe Gram-Schmidt process from v1, v2,...,vk,andletN be the matrix whose rows areu1, u2,...,uk.ThenNN =⎛⎜⎜⎝||u1||20 ... 00 ||u2||2... 0............00... ||uk||2⎞⎟⎟⎠and so det(NN ) is the product of the ||ui||2’s, which is V2.Because each ujis equal to vjminus a linear combination of the preceding vi’s,N = TM where T is a lower triangular matrix with 1’s on its diagonal. Since det(T )=1,det(NN ) = det((TM)(TM) )=det(TMM T )=det(T )det(MM )det(T )=det(MM ). Corollary. The n-dimensional volume of the par allelopiped spanned by the rows of a nby n matrix M is | det(M)|.Proof: By Theorem 5, the volume is det(MM )1/2. Since both matrices are square, thevolume is the square-root ofdet(M)det(M )=det(M)det(M)=(det(M))2,the first equality by Theorem 4. Theorem 6 . If A is a square matri x of order n and S any (reasonable) set in Rn,thenthe n-dimensional volume of the set {Ax : x ∈ S} is | det(A)| times the volume of S.Proof: We’ll talk about this in class. Because the determinant of the transpose of a matrix is the same as the determinantof the matrix, we can infer that the Rules (o)–(iii) hold when the word ‘row’ is replaced by‘column’. So we can understand how the determinant of a matrix changes as elementarycolumn operations are applied. (To add twice column 3 to column 1 of M,wecantakethe transpose, add twice row 3 to row 1, and then transpose again. So the determinantdoes not change when we add twice column 3 to column 1.)One of the reasons that determinants are