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Final - 4 Determinants Lecture 12 Let T R2 R2 be a linear...

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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 defined by det a 11 a 12 a 21 a 22 = fl fl fl fl a 11 a 12 a 21 a 22 fl fl fl fl = a 11 a 22 - a 12 a 21 . It is easy to see that for vectors u , v , w R 2 , we have 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 ] be an n × n square matrix. For a fixed ( 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. Definition 4.1. Let A = [ a ij ] be an n × n square matrix. The determinant of A is a number det A , inductively defined 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 definition 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 = det[ a 11 + a 21 + · · · + a n 1 , a 2 , . . . , a n ] = det[ a 11 , a 2 , . . . , a n ] + det[ a 21 , a 2 , . . . , a n ] + · · · + det[ a n 1 , a 2 , . . . , a n ] = det 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
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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. For any 3 × 3 matrix, fl fl fl fl fl fl a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 fl fl fl fl fl fl = 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 Formula) . 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. fl fl fl fl fl fl fl fl fl a 11 a 12 · · · a 1 n 0 a 22 · · · a 2 n . . . . . . . . . . . . 0 0 · · · a nn fl fl fl fl fl fl fl fl fl = a 11 a 22 · · · a nn . Theorem 4.5. Determinant satisfies 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 ˆ B , then det B = γ det A. Proof. For 2 × 2 case, we have (a) fl fl fl fl a b c + γa d + γb fl fl fl fl = a ( d + γb ) - b ( c + γa ) = ad - bc = fl fl fl fl a b c d fl fl fl fl ; fl fl fl fl a + γc b + γd c d fl fl fl fl = ( a + γc ) d - ( b + γd ) c = ad - bc = fl fl fl fl a b c d fl fl fl fl .
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