同济大学：《线性代数》课程教学资源（课件讲稿）Linear Algebra—Determinants

Linear Algebra-Determinants Li Zhang Department of Mathematics Tongji University 190 Tongji University】 1118

. . Linear Algebra—Determinants Li Zhang Department of Mathematics Tongji University (Tongji University) Linear Algebra 1 / 18

Determinants o Matrices of lower orderd nDuo出 411 21 SeaD世 Tongji University】 2/18

Determinants Matrices of lower order—Diagonal Method . a11. a12. a21. a22. .        .        . = a11a22 . Main Diagonal −a12a21 . Secondary Diag D = . a11. a12. a13. a11. a12. a21. a22. a23. a21. a22. a31. a32. a33. a31. a32. .             .             . − − − + + + D = a11a22a33 + a12a23a31 + a13a21a32 −a11a23a32 − a12a21a33 − a13a22a31 (Tongji University) Linear Algebra 2 / 18

Determinants o Matrices of lower order-Diagonal Method Main Diagonal =2=0 SeaD世 12 Tongji University】 2/18

Determinants Matrices of lower order—Diagonal Method . a11. a12. a21. a22. .        .        . = a11a22 . Main Diagonal −a12a21 . Secondary Diag D = . a11. a12. a13. a11. a12. a21. a22. a23. a21. a22. a31. a32. a33. a31. a32. .             .             . − − − + + + D = a11a22a33 + a12a23a31 + a13a21a32 −a11a23a32 − a12a21a33 − a13a22a31 (Tongji University) Linear Algebra 2 / 18

Determinants o Matrices of lower order-Diagonal Method Main Diagonal a11 a12 Secondary Diag d21 a22 (Tongji University) 2/18

Determinants Matrices of lower order—Diagonal Method . a11. a12. a21. a22. .        .        . = a11a22 . Main Diagonal −a12a21 . Secondary Diag D = . a11. a12. a13. a11. a12. a21. a22. a23. a21. a22. a31. a32. a33. a31. a32. .             .             . − − − + + + D = a11a22a33 + a12a23a31 + a13a21a32 −a11a23a32 − a12a21a33 − a13a22a31 (Tongji University) Linear Algebra 2 / 18

Determinants o Matrices of lower order-Diagonal Method Main Diagonal← d11 a12 =a11a22-2 Secondary Diag d21 Q22 (Tongji University) 2/18

Determinants Matrices of lower order—Diagonal Method . a11. a12. a21. a22. .        .        . = a11a22 . Main Diagonal −a12a21 . Secondary Diag D = . a11. a12. a13. a11. a12. a21. a22. a23. a21. a22. a31. a32. a33. a31. a32. .             .             . − − − + + + D = a11a22a33 + a12a23a31 + a13a21a32 −a11a23a32 − a12a21a33 − a13a22a31 (Tongji University) Linear Algebra 2 / 18

Determinants o Matrices of lower order-Diagonal Method Main Diagonal←a1 a12 =a11a22-a12a21 Secondary Diag+ a21 a22 011 D- 2242 2 12 2 Tongji University】 2/18

Determinants Matrices of lower order—Diagonal Method . a11. a12. a21. a22. .        .        . = a11a22 . Main Diagonal −a12a21 . Secondary Diag D = . a11. a12. a13. a11. a12. a21. a22. a23. a21. a22. a31. a32. a33. a31. a32. .             .             . − − − + + + D = a11a22a33 + a12a23a31 + a13a21a32 −a11a23a32 − a12a21a33 − a13a22a31 (Tongji University) Linear Algebra 2 / 18

Determinants o Matrices of lower order-Diagonal Method Main Diagonal←a1 a12 =a11a22-a12a21 Secondary Diag +a21 Q22 011 D.=222 2 g123 1 2 Tongji University】 2/18

Determinants Matrices of lower order—Diagonal Method . a11. a12. a21. a22. .        .        . = a11a22 . Main Diagonal −a12a21 . Secondary Diag D = . a11. a12. a13. a11. a12. a21. a22. a23. a21. a22. a31. a32. a33. a31. a32. .             .             . − − − + + + D = a11a22a33 + a12a23a31 + a13a21a32 −a11a23a32 − a12a21a33 − a13a22a31 (Tongji University) Linear Algebra 2 / 18

Determinants o Matrices of lower order-Diagonal Method Main Diagonal←a1 a12 =a11a22-a12a21 Secondary Diag a21 Q22 a11 a12 a13 a11 a12 D= a21 422 a23 a21 a22 a31 a32 a33 a31 a32 Tongji University】 2/18

Determinants Matrices of lower order—Diagonal Method . a11. a12. a21. a22. .        .        . = a11a22 . Main Diagonal −a12a21 . Secondary Diag D = . a11. a12. a13. a11. a12. a21. a22. a23. a21. a22. a31. a32. a33. a31. a32. .             .             . − − − + + + D = a11a22a33 + a12a23a31 + a13a21a32 −a11a23a32 − a12a21a33 − a13a22a31 (Tongji University) Linear Algebra 2 / 18

Determinants o Matrices of lower order-Diagonal Method Main Diagonal←a1 a12 =a11a22-a12a21 Secondary Diag+ d21 Q22 a11 a12 a13 a11 a12 D= a21 a22 a23 a21 a22 a31 a32 a33 a31 a32 Tongji University】 2/18

Determinants Matrices of lower order—Diagonal Method . a11. a12. a21. a22. .        .        . = a11a22 . Main Diagonal −a12a21 . Secondary Diag D = . a11. a12. a13. a11. a12. a21. a22. a23. a21. a22. a31. a32. a33. a31. a32. .             .             . − − − + + + D = a11a22a33 + a12a23a31 + a13a21a32 −a11a23a32 − a12a21a33 − a13a22a31 (Tongji University) Linear Algebra 2 / 18

Determinants o Matrices of lower order-Diagonal Method Main Diagonal←a1 a12 =a11a22-a12a21 Secondary Diag+ d21 d22 a11 a12 a13 a11 a12 D= a21 d22 a23 a21 a22 a31 a32 a33 a31 a32 Tongji University】 2/18

Determinants Matrices of lower order—Diagonal Method . a11. a12. a21. a22. .        .        . = a11a22 . Main Diagonal −a12a21 . Secondary Diag D = . a11. a12. a13. a11. a12. a21. a22. a23. a21. a22. a31. a32. a33. a31. a32. .             .             . − − − + + + D = a11a22a33 + a12a23a31 + a13a21a32 −a11a23a32 − a12a21a33 − a13a22a31 (Tongji University) Linear Algebra 2 / 18