Math 317, Tathagata Basak, Some notes on determinant 1 Row operations in terms of matrix multiplication 11 Let I n denote the n n identity matrix Let E ij denote the n n matrix whose (i, j)-th entry is equal to 1 and all other entries are equal to 0 Let A be an n n matrix Performing a row operation on A is equivalent to multiplying A on the left by a certain matrix We can verify this directly by matrix multiplication, once we have written down these matrices Below we write down these matrices (1) Let c be a nonzero scalar Let A be the matrix obtained from A by performing the row operation i c i that multiplies the i-th row of A by c Then A = R I i,c A where RI i,c = I n +(c 1)E ii (2) Let c be a scalar Let A be the matrix obtained by performing the row operation j j + c i that changes the j-th row of A by adding to the j-th row, c times the i-th row of A Then Then A = Ri,j,cA II where Ri,c II = I n + ce ij (3) Let A be the matrix obtained from A by performing the row operation i j that interchanges the i-th and j-th row of A Then A = Ri,j III A where Ri,j III = I n E ii E jj + E ij + E ji Notice also that the inverse operation of each type of elementary row operation above is a row operation of the same kind More precisely, we have ( ) R I 1 ( ) i,c = R I, i, R II 1 ( ) 1 i,j,c = R II i,j, c, R III 1 i,j = R III i,j c 2 Determinant 21 Basic properties of the determinant: Let M n (R) denote the set of all n n matrices To each A M n (R), weshallassociatearealnumber,calledthedeterminantofa, anddenoted det(a) The determinant has the following four properties: (1) If two distinct rows of A are identical, then det(a) =0 (2) Let c be a real number and 1 i n Suppose A, B M n (R) that are identical except in the i-th row and the i-th row of A is c times the i-th row of B Then det(a) =c det(b) (3) Let 1 i n Suppose A, B, C M n (R) that are identical except in the i-th row and the i-th row of A is the sum of the i-th row of B and the i-th row of C Then det(a) = det(b)+det(c) (4) One has det(i n )=1 The following lemma describes the effect of elementary row operations on the determinant In fact these properties hold for any function D : M n (R) R that satisfies the four properties stated above For the rest of the section, we assume that D : M n (R) R is a function satisfying the four properties given in 21 Lemma 22 Let A, B M n (R) Then D also has the following properties: (a) If B is obtained from A by multiplying the i-th row of A by a scalar c, thend(b) =cd(a) In particular, if one of the rows of A is identically 0, thend(a) =0 (b) If B is obtained from A by adding c times i-th row to j-th row for some i j, thend(b) = D(A) (c) If B is obtained from A by interchanging the i-th and j-th row of A for some i j, then D(B) = D(A) 1

Proof Part (a) is simply a restatement of the property (1) For the rest of the proof it will be convenient to write v 1 v 2 A = where v 1 is the first row of A, v 2 is the second row of A and so on; so v j R n for each j So we can think of the determinant as a function from R n R n (n-copies) to R For part (b), we compute: D(B) =D + D + cd (A) v j + c v j c v j v j In the above computation, we have only shown the i-th and j-th rows of each of the matrices Note that except the j-th row, all other rows are identical at each step The second equality follows from property (3), the third uses property (2) and the fourth uses property 1 For part (c) we compute as follows: + v j v j v j v j 0=D +D +D +D +D (A)+D(B) + v j v i v j v i v i v j v i In the above computation, we have only shown the i-th and the j-th row of each of the matrices Note that all the other rows are identical at each step The first and the third equality follows from property (1) and the second equality follows from property (2) applied twice The above lemma has the following immediate corollary Corollary 23 Let A, B M n (R) SupposeB is obtained from A by a finite sequence of elementary row operations It is understood here that for each row operation of the form i c i, the scalar c is nonzero (a) One has det(a) =D(A) if and only if det(b) =D(B) (b) One has D(B) =0if and only if D(A) =0 Proof (a) From the previous lemma we see that each elementary row operation changes both det(a) and D(A) byexactlythesamefactorpart(a)follows (b) The previous lemma also implies that each elementary row operation changes D(A) bya nonzero factor Part (b) follows from this Lemma 24 Let A M n (R) is upper triangular Then D(A) is the product of the diagonal entries of A Proof Let a ij be the (i, j)-th entry of A First suppose some diagonal entry of A is equal to 0 Choose the largest i such that the i-th diagonal entry is equal to 0 If i = n, thenthen-th row of 2 v n

A is identically 0, so D(A) =0 Ifi<n,thennotethata i+1,i+1,,a n,n are all nonzero Let B be the matrix obtained from A after performing the successive row operations i i a i,n a n,n n i i a i,n 1 a n 1,n 1 n 1 i i a i,i+1 a i+1,i+1 i +1 One verifies that the i-th row of B is identically 0 So D(B) =0 Nowpart(b)oftheprevious lemma tells us that D(A) =0aswell ThuswehavearguedthatifsomediagonalentryofA is zero, then D(A) = 0 SonowassumethatallthediagonalentriesofA are nonzero Then we can perform row operations of type II to change A to a matrix B =diag(a 11,a 22,,a nn ) From Lemma 22 (b), we know that row operations of type II does not change the value of D, so D(A) =D(B) =a 11 a 22 a nn Theorem 25 There is a unique function det : M n (R) R satisfying the four properties in 21 plan of proof We shall omit the details proof of existence but here is how onemayproceed Define determinants explicitly as follows: Let A be an n n matrix whose (i, j)-th entry is a ij LetA ij be the (n 1) (n 1) matrix obtained from A by deleting the i-th row and the j-th column Then the determinant can be defined inductively by: n (1) det(a) = ( 1) i+j a ij det(a ij ) for any i =1,,n j=1 Now one needs to show that this function satisfies the four properties given in 21 The main work is needed to show that the right hand side of equation (1) has the same value for each i Then it quickly follows that interchanging two rows of a matrix changes the determinant by a sign Verifying the other properties are then easy proof of uniqueness: Suppose D : M n (R) R is some function satisfying the four properties given in 21 Let A be any n n matrix We need to argue that D(A) =det(a) We know that we can start with A and apply a finite number of elementary row operations to obtain a matrix B that is upper triangular Lemma 24 implies that for an upper triangular matrix D(B) = det(b) since both sides are equal to the product of the diagnonal entries of B Nowlemma23(a)implies that D(A) = det(a) We end stating a couple of more theorems We shall omit the proofs Theorem 26 Let A be an n n matrices Then the following are equivalent: (a) A is invertible (b) rank(a) =n (c) det(a) 0 Theorem 27 Let A and B be n n matrices Then det(ab) =det(a)det(b) 3