Matrix algebra determining errors
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The document discusses matrix algebra concepts like determining the determinant of a 3x3 matrix using different methods. It provides an example of a 3x3 matrix and calculates its determinant using criss-cross multiplication, getting -15. It then checks for errors by calculating the determinant using row and column methods, finding inconsistencies. A tip is provided that the error occurs in the middle row/column if criss-cross was correct, or first/last if criss-cross was incorrect, to help identify the right determinant value.
Matrix algebra determining errors
- 2. A systematic approach of the elimination method for solving a system of linear equations provides another method of solution that involves a simplified notation. 3 ways in finding determinants: Criss-cross multiplication Row Column
- 3. DETERMINING THE ERROR OF 3X3 MATRIX
- 4. The Given Matrix: 3 1 1 A= 2 -4 -3 7 -2 0
- 5. 3 1 1 3 1 2 -4 -3 2 -4 7 -2 0 7 -2 (0 -21 -4) - (-28 +18+0 ) = -15 Criss-cross multiplication
- 6. Cofactor: 3= -4 -3 -2 0 = -6 1= 2 -3 7 0 = 21 1= 2 -4 7 -2 = 24
- 7. Cofactor: 2= 1 1 -2 0 =2 -4= 3 1 7 0 = -7 -3= 3 1 7 -2 = -13
- 8. Cofactor: 7= 1 1 -4 -3 =1 -2= 3 1 2 -3 = -11 0= 3 1 2 -4 = -14
- 9. Inverse Matrix: A-1 = -1/15 -6 2 1 + - + 21 -7 -11 - + - 24 -13 -14 + - + 6/15 2/15 -1/15 A-1 = 21/15 7/15 -11/15 -24/15 -13/15 14/15
- 10. Identity Matrix 3 1 1 6/15 2/15 -1/15 AA-1= 2 -4 -3 21/15 7/15 -11/15 7 -2 0 -24/15 -13/15 14/15 1 0 0 = 0 1 0 0 0 1
- 11. Remember: • The first thing we should do is to identify the correct determinant and finding the inverse and identity of the matrix given was done in order to prove whether the determinant used wasn’t wrong.
- 12. ERRORS Criss-Cross Multiplication Row Determinant Column Determinant
- 13. Criss-Cross 3 1 1 3 1 2 -4 -3 2 4 7 -2 0 7 -2 = -21 - 4 + 28 – 18 = -15
- 14. Criss-Cross 7 -2 0 7 -2 3 1 1 3 1 2 -4 -3 2 -4 = -21 - 4 + 28 – 18 = -15
- 15. Criss-Cross 7 -2 0 7 -2 2 -4 -3 2 -4 3 1 1 3 1 = -28 + 18 + 21 + 4 = 15 ERROR
- 16. Criss-Cross 3 1 1 3 1 7 -2 0 7 -2 2 -4 -3 2 -4 = 18 - 28 + 4 + 21 = 15 ERROR
- 17. Criss-Cross 2 -4 -3 2 -4 3 1 1 3 1 7 -2 0 7 -2 = -28 +18 + 21 – 4 = 15 ERROR
- 18. Criss-Cross 2 -4 -3 2 4 7 -2 0 7 -2 3 1 1 3 1 = -4 - 21 - 18 + 28 = -15
- 19. Column 3 1 1 2 -4 -3 7 -2 0 = 1(24) + 3(-13) + 0 = -15 = 1(21) + 4(-7) - 2(-11) = 15 ERROR = 3(-6) – 2(2) + 7(1) = -15
- 20. Column 3 1 1 7 -2 0 2 -4 -3 = 1(-24) – 0 – 3(-13) = 15 ERROR = 1(-21) + 2(-11) - 4(-7) = -15 = 3(6) – 7(1) + 2(2) = 15 ERROR
- 21. ROW 3 1 1 2 -4 -3 7 -2 0 = 7(1) + 2(-11) + 0(-14) = -15 = 2(2) + 4(-7) - 3(-13) = 15 ERROR = 3(-6) – 1(21) + 1(24) = -15
- 22. ROW 3 1 1 7 -2 0 2 -4 -3 = 2(2) + 4(-7) – 3(-13) = 15 ERROR = 7(1) + 2(-11) - O(-14) = -15 = 3(6) – 1(-21) + 1(-24) = 15 ERROR
- 23. Tip in finding the error: If the determinant you’ve found using criss-cross multiplication in matrix given is correct, the error in row and column was found in the middle row and column but if the determinant you’ve found using criss-cross multiplication in the given matrix is the error, the error in row and column was found in the first and last row and column. •