D. Determinants. a b c d e f

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1 D. Determinants Given a square array of numers, we associate with it a numer called the determinant of, and written either det(), or. For 2 2 and 3 3 arrays, the numer is defined y () a c d = ad c; a c d e f = aei+fg +dhc ceg di fha. g h i Do not memorize these as formulas learn instead the patterns which give the terms. The 2 2 case is easy: the product of the elements on one diagonal (the main diagonal ), minus the product of the elements on the other (the antidiagonal ). For the 3 3 case, three products get the + sign: those formed from the main diagonal, or having two factors parallel to the main diagonal. The other three get a negative sign: those from the antidiagonal, or having two factors parallel to it. Try the following example on your own, then check your work against the solution. 2 Example. Evaluate 3 2 using (). 2 4 Solution. Using the same order as in (), we get 2+( 8)+ 6 8 ( 2) = 7. Important facts aout : D-. is multiplied y if we interchange two rows or two columns. D-2. = 0 if one row or column is all zero, or if two rows or two columns are the same. D-3. is multiplied y c, if every element of some row or column is multiplied y c. D-4. The value of is unchanged if we add to one row (or column) a constant multiple of another row (resp. column). ll of these facts are easy to check for 2 2 determinants from the formula (); from this, their truth also for 3 3 determinants will follow from the Laplace expansion. Though the letters a,,c,... can e used for very small determinants, they can t for larger ones; it s important early on to get used to the standard notation for the entries of determinants. This is what the common software packages and the literature use. The determinants of order two and three would e written respectively a a 2 a a 2 a 3 a 2 a 22 a 2 a 22 a 23 a 3 a 32 a 33 There is another form for this rule which requires adding two extra columns to the determinant, ut this wastes too much time in practice and leads to awkward write-ups; instead, learn to evaluate each of the six products mentally, writing it down with the correct sign, and then add the six numers, as is done in Example. Note that the word determinant is also used for the square array itself, enclosed etween two vertical lines, as for example when one speaks of the second row of the determinant.

2 NOTES In general, the ij-entry, written a ij, is the numer in the i-th row and j-th column. Its ij-minor, written ij, is the determinant that s left after deleting from the row and column containing a ij. Its ij-cofactor, written here ij, is given as a formula y ij = ( ) i+j ij. For a 3 3 determinant, it is easier to think of it this way: we put + or in front of the ij-minor, according to whether + or occurs in the ij-position in the checkeroard pattern + + (2) Example.2 = 2 2. Find 2, 2, 22, 22. Solution. 2 = 2 =, 2 =. 22 = 3 2 = 7, 22 = 7. Laplace expansion y cofactors This is another way to evaluate a determinant; we give the rule for a 3 3. It generalizes easily to an n n determinant. Select any row (or column) of the determinant. Multiply each entry a ij in that row (or column) y its cofactor ij, and add the three resulting numers; you get the value of the determinant. s practice with notation, here is the formula for the Laplace expansion of a third order (i.e., a 3 3) determinant using the cofactors of the first row: (3) a +a 2 2 +a 3 3 = and the formula using the cofactors of the j-th column: (4) a j j +a 2j 2j +a 3j 3j = Example.3 Evaluate the determinant in Example.2 using the Laplace expansions y the first row and y the second column, and check y also using (). Solution. The Laplace expansion y the first row is = = ( ) 0 +3 ( 3) = 0. The Laplace expansion y the second column would e = = 0+2 ( 7) ( 4) = 0.

3 D. DETERMINNTS 3 Checking y (), we have = ( ) = 0. Example.4 Show the Laplace expansion y the first row agrees with definition (). Solution. We have a c d e f g h i = a e f d f +c d e h i g i g h = a(ei fh) (di fg)+c(dh eg), whose six terms agree with the six terms on the right of definition (). ( similar argument can e made for the Laplace expansion y any row or column.) rea and volume interpretation of the determinant: (5) ± a a 2 2 = area of parallelogram with edges = (a,a 2 ), = (, 2 ). a a 2 a 3 (6) ± 2 3 c c 2 c 3 = volume of parallelepiped with edges row-vectors,,c. C In each case, choose the sign which makes the left side non-negative. Proof of (5). We egin with two preliminary oservations. Let e the positive angle from to ; we assume it is < π, so that and have the general positions illustrated. Let = π/2, as illustrated. Then cos = sin. Draw the vector otained y rotating to the right y π/2. The picture shows that = ( 2, ), and =. To prove (5) now, we have a standard formula of Euclidean geometry, area of parallelogram = sin = cos, y the aove oservations =, y the geometric definition of dot product = a 2 a 2 y the formula for 2 2 This proves the area interpretation (5) if and have the position shown. If their positions are reversed, then the area is the same, ut the sign of the determinant is changed, so the formula has to read, area of parallelogram = ± a a 2 2, whichever sign makes the right side 0. The proof of the analogous volume formula (6) will e made when we study the scalar triple product C.

4 NOTES For n n determinants, the analog of definition () is a it complicated, and not used to compute them; that s done y the analog of the Laplace expansion, which we give in a moment, or y using Fact D-4 in a systematic way to make the entries elow the main diagonal all zero. Generalizing (5) and (6), n n determinants can e interpreted as the hypervolume in n-space of a n-dimensional parallelotope. For n n determinants, the minor ij of the entry a ij is defined to e the determinant otained y deleting the i-th row and j-th column; the cofactor ij is the minor, prefixed y a + or sign according to the natural generalization of the checkeroard pattern (2). Then the Laplace expansion y the i-th row would e = a i i +a i2 i a in in. This is an inductive calculation it expresses the determinant of order n in terms of determinants of order n. Thus, since we can calculate determinants of order 3, it allows us to calculate determinants of order 4; then determinants of order 5, and so on. If we take for definiteness i =, then the aove Laplace expansion formula can e used as the asis of an inductive definition of the n n determinant Example.5 Evaluate y its Laplace expansion y the first row Solution = ( 23) 3 2 = 3. Exercises: Section C

5 8.02 Notes and Exercises y. Mattuck and jorn Poonen with the assistance of T.Shifrin and S. LeDuc c M.I.T