lach4 - D ETERMINANTS There a re s everal w ays t o d efine...

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DETERMINANTS There are several ways to define the determinant of a square rnatrix. In second semester calculus you would have encountered dcterminants of 2 x 2 arrd 3 x 3 matrices. For a 2 x 2 matrix A: (o1,i) the cleterrninant of A is defined as det A: a11a22 - a2tat.z. Anotlrer notation for det A is lAl. For examole. To calculate 3 x 3 you probably usetl the diagonal rule : 2(-7) _ 3(5) : _74 _ 75 : _29. a11a220gg I apl23a3t I a13A21o,32 ot: o -/ Att AtZ a, t: A2t 022 A23 u3t a3Z ASt a31A22clg A32a2ga12 A93a210"12. This is called the diagonal nrle since c:rch of terms above r:orrespond to entries on the indicated rliagonals given bclow: ::>:">::x:,1: ";aa7"J1,;,\;. rule, we havc For example, by t cliagonal :)r J -i) -1 2 4-I 2(-1)( 1) + (3)(2)(3) F ( 5)(1)(4) (3)(-1)( 5) (4)(2)(2) (-1)(1)(3) :2+18-20-15 16+3 t11 Warning: The detcrrninant of a 4 x 4 rnal,rix r;iln not be calculated by drawing diagonals. To calculate higher order determinants two rrrethotls are used: (1) cofactor expansions, (2) clemerttary operations. The textbook defines the general n x n cleterrninzrnt via cofactor expansions. See the definition on page 104. he 2 1 ) t) Given an n xn matrix A: (a;i). Let Mai be the (n 1) x deleting the f-th row and j-th colum' of A. Then the cofactor row is rL cler A t aii(t1i+i 4ct Mai. j:t (n 1) rnatrix gotten blr expansion alorrg the i-th
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Also the cofactor expansion along the j-th column is detA : t ai1(t)i+i 4et Mii. r-l ExamPlc'Let /r 2 3\ A:lo 5 o l. \5 -1 *2/ (i) Calculate det A using a cofactor exparrsion alorrg the second row. (ii) Calculate dct,4 using a cofactor expansion along the third column. (i) BV a cofactor expansion along the second row wc have det,4 : -0 - 5(- (ii) By a cofacl,or cxparrsi<-rrr det ,4 : .10 ol- ID 5(-17) : -85. :rlons tirc third t;olurnn wc havc .t .) LO ''| o -l L o 1r,\ - L \O ) - :3(0- *0- 72 5 -1 2 10 rr fJ I2 2(5 - 13 5-2 0 + (-2) \-. nf. t') -u or.\ Lo ) -U l; To calculate the detcrminant using elerrrentary operations wc nced the following prolF ert;ies of the deterrninant: (1) The effect on thc determinant by a Type I opcration (swap two rows or two columrrs) is to change the sign of the determina"nt.
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lach4 - D ETERMINANTS There a re s everal w ays t o d efine...

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