Math 307 Spring, 2005 Hentzel Time: 10:00 to 10:50 MWF Room: 205 Carver Hall Instructor: Irvin Roy Hentzel Office 432 Carver Phone 515-294-8141 E-mail: hentzel@iastate.edu http://orion.math.iastate.edu/hentzel/class.307.05 Text: Linear Algebra and its Applications, Third Edition David C. Lay Friday, February 25 3.1 Assignment Page 191 Problem 18 through 30. Main Idea: There is a magical number associated to a matrix called the determinant. It represents how much the matrix changes volumes. Key Words: Determinants, Elementary Row Operations, sgn Det[A] = SUM sgn(p) a a ... a all p 1 p(1) 2 p(2) n p(n) Goal: Learn the definition of a determinant. Learn how the elementary row operations affect the determinant. Learn det(A B) = det(A) det(B) ============================================================== Today we do the following. 1. Give the definition of the determinant of Anxn 2. Show how the elementary row operations affect the determinant. 3. Show that det[A B] = det[A] det[B]. ---------------------------------------------------------------- We give the definition in English. Given a nxn matrix A, (a) Pick n elements of A no two of which are in the same row or column. (b) Multiply then together. (c) Calculate the appropriate sign. (d) Do this for all possible choices of n elements. (e) The sum of these terms is the determinant of A. ------------------------------------------------------------- We give the definition using summation notation. Det[A] = SUM sgn(p) a a ... a all p 1 p(1) 2 p(2) n p(n) Since the n elements are from n different rows, one is in the first row, which is a1 p(1); one is from the second row, which is a2 p(2); etc. The calculated sign is indicated by sgn(p). It depends on the second coordinates of the chosen elements. --------------------------------------------------------------------- Observation: If you start with a row of objects in order, and you start switching objects, two at a time. It will always take an even number of such switches to return the row of objects to their original order. There are two distinct classes of arrangements: those obtained by an even number of switches, and those obtained by an odd number of switches. switching two elements moves the arrangement from one side to another. ____________________ ________________________ | | | | | | | | | 12345 | | 12354 | | 0 inversion | | 1 inversion | | | | | | 12534 | | 15234 | | 2 inversions | | 3 inversion | | | | | | etc. | | etc. | |__________________| |______________________| It is pretty clear that adjacent interchanges will either increase the number of inversions by one, or decrease the number of inversions by one. And thus will change the number of inversions from odd to even, or from even to odd. Since any interchange can be accomplished by a series of adjacent interchanges, the interchange of any two elements, whether adjacent or not, will change the parity. Those from an odd number of switches are called odd permutations. Those from an even number of switches are called even permutations. It is easy to determine if a permutation is even or odd by counting the number of "inversions". That is, the number of times a larger number precedes a smaller number. The sgn is +1 if the number of inversions is even. The sgn is -1 if the number of inversions is odd. ------------------------------------------------------------ Example: a14 a23 a32 a41 look at 4 3 2 1 there are six instances where a larger precedes a smaller. the sgn is +1. ------------------------------------------------------------ Example: a11 a23 a32 a44 look at 1324 there is one instance where a larger number precedes a smaller. the sgn is -1. Example: | a11 a12 a13 | +a11 a22 a33 -a13 a22 a31 | a21 a22 a23 | +a12 a23 a31 -a11 a23 a32 | a31 a32 a33 | +a13 a21 a32 -a12 a21 a33 ================================================================= Now we look at how the elementary row operations affect the determinant. (1) Switching two rows changes the sign of the determinant. (2) Multiply a row by a scalar c multiplies the determinant by c. (3) Adding a multiple of one row to another does not change the determinant. ------------------------------------------------------------------ Elementary Row Operation (1). When the rows i and j are switched, the sign is computed with the positions p(i) and p(j) interchanged. Thus the sgn of each term is switched, so the sign of the whole determinant changes. ============================================================== Elementary Row Operation (2) Multiplying a row by a scalar c multiplies the determinant by c. | ---R1 --- | | --- R1 ----| | ---R2 --- | | --- R2 ----| | --- --- | | --- ----| Det| c Ri | = c Det | Ri | | --- --- | | --- ----| | --- --- | | --- ----| | ---Rn --- | | --- Rn ----| ----------------------------------------------------------- If we multiply the elements of a row by c, there appears exactly one c in each of the terms. We can factor the c out and place it before the summation. ----------------------------------------------------------------- If A' is A with row i multiplied by c then Det[A'] = SUM sgn(p) a1 p(1) a2 p(2) ... (c ai p(i)).... an p(n). all p = c SUM sgn(p) a1 p(1) a2 p(2) ... ai p(i).... an p(n). all p = c Det[A]. ==================================================================== Elementary Row Operation (3). Adding a multiple of one row to another does not change the determinant. | ---R1 --- | | --- R1 ----| | --- R1 ----| | Ri + c Rj | | --- Ri ----| | --c Rj ----| | --- --- | | --- ----| | --- ----| Det| --- --- | = | --- ----| + | --- ----| | ---Rj --- | | --- Rj ----| | --- Rj ----| | --- --- | | --- ----| | --- ----| | ---Rn --- | | --- Rn ----| | --- Rn ----| This is true because each term in the summation has one entry from Ri + c Rj. The term can be expanded into two pieces. Put the part from Ri into the first and the part from c Rj into the second. | --- R1 ----| | --- R1 ----| | --- Ri ----| | -- Rj ----| | --- ----| | --- ----| = | --- ----| +c | --- ----| | --- Rj ----| | --- Rj ----| | --- ----| | --- ----| | --- Rn ----| | --- Rn ----| The c can be brought outside by using Elementary Row Operation (2) | --- R1 ----| | --- Ri ----| | --- ----| = | --- ----| + 0 | --- Rj ----| | --- ----| | --- Rn ----| The determinant of a matrix with two identical rows has to be zero. If we switch the identical rows, the sign of the determinant must change by Elementary Row Operation (1). But since the rows were identical, the determinant has to be the same. This Det(A) = - Det(A) only happens when Det(A) = 0. ================================================================= We can summarize the above results in this way. If E is an Elementary Row Operation Matrix, then Det[E A] = Det[E] Det[A] for any matrix A nxn. ================================================================ Theorem: A is invertible <==> Det[A] =/= 0. Proof: A is invertible <==> RCF(A) = I. If RCF(A) = I, then Det[ RCF(A) ] = 1. If RCF(A) =/= I, then Det[ RCF(A) ] = 0 since RCF(A) has a row of zeros. Thus A is invertible <==> Det[ RCF(A) ] =/= 0. Reduce A to row canonical form using elementary row operations. Ek Ek-1 ... E3 E2 E1 A = RCF(A). Using the fact that Det[ E A ] = Det[E] Det[A] we get Det[Ek] Det[Ek-1] ... Det[E3] Det[E2] Det[E1] Det[A] = Det[ RCF(A) ]. Since each of the factors Det[ Ei ] is nonzero, we know that Det[A] = 0 <===> Det[ RCF(A) ] = 0. Combine this with the first statement says that A is invertible <==> Det[A] =/= 0. ------------------------------------------------------------------- Theorem: Det[ A B ] = Det[A] Det[B]. Proof. If A is not invertible, then A B is not invertible and so Det[A] = 0, Det[A B] = 0, and hence Det[A B ] = Det[A] Det[B]. If A is invertible, then A is a product of Elementary Row Operation Matrices. A = E1 E2 ... Ek. Det[ A B ] = Det[ E1 E2 ... Ek B] = Det[E1] Det[E2] ... Det[Ek] Det[B] = Det[E1 E2 ... Ek] Det[B] = Det[A] Det[B].