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Home > Matrix & Vector calculators > value of determinant using properties of determinants example
21. determinants using properties of determinants example ( Enter your problem )
( Enter your problem )
  1. Example `[[201,210,220],[151,155,140],[50,55,80]]`
  2. Example `[[100,205,105],[200,408,207],[300,608,310]]`
  3. Example `[[2,1970,1978],[5,1960,1980],[7,1950,1978]]`
  4. Example `[[1977,1979,1981],[1940,1943,1946],[10,17,24]]`
 		Other related methods
  1. Transforming matrix to Row Echelon Form
  2. Transforming matrix to Reduced Row Echelon Form
  3. Rank of matrix
  4. Characteristic polynomial of matrix
  5. Eigenvalues
  6. Eigenvectors (Eigenspace)
  7. Triangular Matrix
  8. LU decomposition using Gauss Elimination method of matrix
  9. LU decomposition using Doolittle's method of matrix
  10. LU decomposition using Crout's method of matrix
  11. Diagonal Matrix
  12. Cholesky Decomposition
  13. QR Decomposition (Gram Schmidt Method)
  14. QR Decomposition (Householder Method)
  15. LQ Decomposition
  16. Pivots
  17. Singular Value Decomposition (SVD)
  18. Moore-Penrose Pseudoinverse
  19. Power Method for dominant eigenvalue
  20. determinants using Sarrus Rule
  21. determinants using properties of determinants
  22. Row Space
  23. Column Space
  24. Null Space
2. Example `[[100,205,105],[200,408,207],[300,608,310]]`
(Previous example)		4. Example `[[1977,1979,1981],[1940,1943,1946],[10,17,24]]`
(Next example)

3. Example `[[2,1970,1978],[5,1960,1980],[7,1950,1978]]`


Find value of determinant using properties of determinants ...
`[[2,1970,1978],[5,1960,1980],[7,1950,1978]]`

Solution:
 `A=` 
 2  1970  1978 
 5  1960  1980 
 7  1950  1978 


Now, `C_3=C_3 - C_2`

 `=` 
 2  1970  8 
 5  1960  20 
 7  1950  28 


take 4 as a comman factor from `C_3`

 `=4 xx ` 
 2  1970  2 
 5  1960  5 
 7  1950  7 


Here `C_1=C_3`, So value of the determinant is 0

`=4 xx 0`

`=0`

Method-2: Determinant by expanding cofactors

`|A|` = 
 `2`  `1970`  `1978` 
 `5`  `1960`  `1980` 
 `7`  `1950`  `1978` 


 =
 `2` × 
 `1960`  `1980` 
 `1950`  `1978` 
 `-1970` × 
 `5`  `1980` 
 `7`  `1978` 
 `+1978` × 
 `5`  `1960` 
 `7`  `1950` 


`=2 xx (1960 × 1978 - 1980 × 1950) -1970 xx (5 × 1978 - 1980 × 7) +1978 xx (5 × 1950 - 1960 × 7)`

`=2 xx (3876880 -3861000) -1970 xx (9890 -13860) +1978 xx (9750 -13720)`

`=2 xx (15880) -1970 xx (-3970) +1978 xx (-3970)`

`= 31760 +7820900 -7852660`

`=0`



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2. Example `[[100,205,105],[200,408,207],[300,608,310]]`
(Previous example)		4. Example `[[1977,1979,1981],[1940,1943,1946],[10,17,24]]`
(Next example)


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