Final42 - (b) cd ab ab c d = cb da = (ad bc) = ab . cd ab ....

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4 Determinants Lecture 12 Let T : R 2 R 2 be a linear transformation with the standard matrix A = · a 11 a 12 a 21 a 22 ¸ . Recall that the determinant of a 2 × 2 matrix h a 11 a 12 a 21 a 22 i is deFned by det · a 11 a 12 a 21 a 22 ¸ = ¯ ¯ ¯ ¯ a 11 a 12 a 21 a 22 ¯ ¯ ¯ ¯ = a 11 a 22 a 12 a 21 . It is easy to see that for vectors u , v , w R 2 ,wehave det[ a u + b v , w ]= a det[ u , w ]+ b det[ v , w ] , det[ u ,a v + b w a det[ u , v b det[ u , w ] . This means that det is linear in each column vector variables. Moreover, switching two columns (rows) changes the sign, i.e., det[ u , v det[ v , u ] . Let A =[ a ij ]bean n × n square matrix. ±or a Fxed ( i, j ), where 1 i m and 1 j n ,let A ij =( n 1) × ( n 1) submatrix of A obtained by deleting the i th row and j th column of A. Defnition 4.1. Let A a ij n × n square matrix. The determinant of A is a number det A , inductively deFned by det A = a 11 det A 11 a 12 det A 12 + ··· +( 1) n +1 a 1 n det A 1 n = a 11 det A 11 a 21 det A 21 + 1) n +1 a n 1 det A n 1 . The motivation of this deFnition is as follows: Write A a 1 , a 2 , , a n ], Note that a 1 = a 11 + a 21 + + a n 1 ,where a 11 = a 11 0 . . . 0 , a 21 = 0 a 21 . . . 0 , a n 1 = 0 0 . . . a n 1 . Since det A is linear in each column vectors of the matrix A , i.e., det[ a u + b v , a 2 ,..., a n a det[ u , a 2 a n b det[ v , a 2 a n ], we have det A =d e t [ a 11 + a 21 + + a n 1 , a 2 a n ] e t [ a 11 , a 2 a n ]+det[ a 21 , a 2 a n +det[ a n 1 , a 2 a n ] e t · a 11 0 A 11 ¸ det · a 21 0 A 21 ¸ + 1) n 1 det · a n 1 0 A n 1 ¸ = a 11 det A 11 a 21 det A 21 + 1) n +1 a n 1 det A n 1 . 15 Theorem 4.2. Let A a ij ] be an n × n matrix. The ( i, j ) -coFactor of A (1 i, j n ) is the number C ij 1) i + j det A ij . Then det A = a 11 C 11 + a 12 C 12 + + a 1 n C 1 n = a 11 C 11 + a 21 C 21 + + a n 1 C n 1 . Example 4.1. ±or any 3 × 3 matrix, ¯ ¯ ¯ ¯ ¯ ¯ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ¯ ¯ ¯ ¯ ¯ ¯ = a 11 a 22 a 33 + a 12 a 23 a 31 + a 13 a 21 a 32 a 11 a 23 a 32 a 12 a 21 a 33 a 13 a 22 a 31 . Theorem 4.3 (Cofactor Expansion ±ormula) . For any n × n square matrix A a ij ] , det A = a i 1 C i 1 + a i 2 C i 2 + + a in C in = a 1 j C 1 j + a 2 j C 2 j + + a nj C nj . In other words, det A T =det A. Proposition 4.4. ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ a 11 a 12 a 1 n 0 a 22 a 2 n . . . . . . . . . . . . 00 a nn ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ = a 11 a 22 a nn . Theorem 4.5. Determinant satis±es the following properties. (a) Adding a multiple of one row (column) to another row (column) does not change the deter- minant. (b) Interchanging two rows (columns) changes the sign of the determinant. (c) If two rows (columns) are the same, then the determinant is zero. (d) If one row (column) of A is multiplied by a scalar γ to produce a matrix B , i.e., A γR k à B , then det B = γ det A. Proof. ±or 2 × 2 case, we have (a) ¯ ¯ ¯ ¯ ab c + γa d + γb ¯ ¯ ¯ ¯ = a ( d + ) b ( c + ) = ad bc = ¯ ¯ ¯ ¯ cd ¯ ¯ ¯ ¯ ; ¯ ¯ ¯ ¯ a + γc b + γd ¯ ¯ ¯ ¯ a + ) d ( b + ) c = ad bc = ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ .
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Final42 - (b) cd ab ab c d = cb da = (ad bc) = ab . cd ab ....

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