同济大学：《线性代数》课程教学资源（课件讲稿）Determinants

Another Way 11 Tongji University】 6/32

. Another Way D = . a11. a12. a13. a11. a12. a21. a22. a23. a21. a22. a31. a32. a33. a31. a32. .             .             . − − − + + + D = a11a22a33 + a12a23a31 + a13a21a32 −a11a23a32 − a12a21a33 − a13a22a31 . Note: . . The plus signs and the minus signs The rule doesn’t work for higher matrix There are 3! elementary products. (Tongji University) Linear Algebra 6 / 32

Another Way a11 a12 a13 a11 412 D= a21 122 a23 21 a22 a31 032 a33 31 a32 (Tongji University) 6/32

. Another Way D = . a11. a12. a13. a11. a12. a21. a22. a23. a21. a22. a31. a32. a33. a31. a32. .             .             . − − − + + + D = a11a22a33 + a12a23a31 + a13a21a32 −a11a23a32 − a12a21a33 − a13a22a31 . Note: . . The plus signs and the minus signs The rule doesn’t work for higher matrix There are 3! elementary products. (Tongji University) Linear Algebra 6 / 32

Another Way a11 Q12 a13 a11 412 D= a21 22 a23 21 a22 a31 32 a33 a31 a32 (Tongji University) 6/32

. Another Way D = . a11. a12. a13. a11. a12. a21. a22. a23. a21. a22. a31. a32. a33. a31. a32. .             .             . − − − + + + D = a11a22a33 + a12a23a31 + a13a21a32 −a11a23a32 − a12a21a33 − a13a22a31 . Note: . . The plus signs and the minus signs The rule doesn’t work for higher matrix There are 3! elementary products. (Tongji University) Linear Algebra 6 / 32

Another Way a11 a12 a13 a11 412 D= a21 22 a23 21 a22 a31 a32 a33 a31 a32 y + + (Tongji University) 6/32

. Another Way D = . a11. a12. a13. a11. a12. a21. a22. a23. a21. a22. a31. a32. a33. a31. a32. .             .             . − − − + + + D = a11a22a33 + a12a23a31 + a13a21a32 −a11a23a32 − a12a21a33 − a13a22a31 . Note: . . The plus signs and the minus signs The rule doesn’t work for higher matrix There are 3! elementary products. (Tongji University) Linear Algebra 6 / 32

Another Way a11 a12 a13 a11 412 D= a21 22 a23 a21 a22 a31 a32 a33 a31 a32 ￥￥ + (Tongji University) 6/32

. Another Way D = . a11. a12. a13. a11. a12. a21. a22. a23. a21. a22. a31. a32. a33. a31. a32. .             .             . − − − + + + D = a11a22a33 + a12a23a31 + a13a21a32 −a11a23a32 − a12a21a33 − a13a22a31 . Note: . . The plus signs and the minus signs The rule doesn’t work for higher matrix There are 3! elementary products. (Tongji University) Linear Algebra 6 / 32