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Understanding Determinants and Cramer’s Rule for Solving Linear Equations

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This guide provides essential information on determinants of square matrices, focusing on 2x2 and 3x3 matrices. The determinant for a 2x2 matrix is calculated as det.A = ad - bc, and for a 3x3 matrix, it involves a more complex formula: det.A = (aei + bfg + cdh) - (ceg + afh + bdi). Additionally, we explore Cramer’s Rule, which utilizes determinants to solve systems of linear equations, exemplified through specific equations. Practice problems are included to reinforce understanding.

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Understanding Determinants and Cramer’s Rule for Solving Linear Equations

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  1. Determinants Notation: det A |A| Can only be done for SQUARE matrices. (2x2, 3x3, etc.)

  2. Determinant for 2x2 Matrix det A = ad - bc

  3. Determinant for 2x2 Matrix det A = ad - bc

  4. Determinant for 2x2 Matrix det A = ad - bc det C = 5(3) – 8(2)

  5. Determinant for 2x2 Matrix det A = ad - bc det C = 5(3) – 8(2) = 15 – 16 = -1

  6. Determinant for 3x3 Matrix

  7. Determinant for 3x3 Matrix

  8. Determinant for 3x3 Matrix det A = (aei + bfg + cdh)

  9. Determinant for 3x3 Matrix det A = (aei + bfg + cdh) – (ceg + afh + bdi)

  10. Determinant for 3x3 Matrix det A = (aei + bfg + cdh) – (ceg + afh + bdi) det B =

  11. Determinant for 3x3 Matrix det A = (aei + bfg + cdh) – (ceg + afh + bdi) det B = (1*4*9 + 0*1*6 + 5*2*3)

  12. Determinant for 3x3 Matrix det A = (aei + bfg + cdh) – (ceg + afh + bdi) det B = (1*4*9 + 0*1*6 + 5*2*3) – (6*4*5 + 3*1*1 + 9*2*0)

  13. Determinant for 3x3 Matrix det A = (aei + bfg + cdh) – (ceg + afh + bdi) det B = (1*4*9 + 0*1*6 + 5*2*3) – (6*4*5 + 3*1*1 + 9*2*0) = (36 + 0 + 30) – (120 + 3 + 0)

  14. Determinant for 3x3 Matrix det A = (aei + bfg + cdh) – (ceg + afh + bdi) det B = (1*4*9 + 0*1*6 + 5*2*3) – (6*4*5 + 3*1*1 + 9*2*0) = (36 + 0 + 30) – (120 + 3 + 0) = 66 – 123 = -57

  15. Cramer’s Rule Uses matrices to solve systems of equations. ax + by = e cx + dy = f

  16. Cramer’s Rule Uses matrices to solve systems of equations. ax + by = e cx + dy = f Coefficient Matrix

  17. Cramer’s Rule Uses matrices to solve systems of equations. ax + by = e cx + dy = f Coefficient Matrix

  18. Cramer’s Rule Uses matrices to solve systems of equations. ax + by = e cx + dy = f Coefficient Matrix

  19. Cramer’s Rule Uses matrices to solve systems of equations. ax + by = e cx + dy = f Coefficient Matrix

  20. Cramer’s Rule Uses matrices to solve systems of equations. ax + by = e cx + dy = f Coefficient Matrix

  21. Cramer’s Rule 4x – 2y = -6 -3x + y = -3

  22. Cramer’s Rule 4x – 2y = -6 -3x + y = -3 Coefficient Matrix

  23. Cramer’s Rule 4x – 2y = -6 -3x + y = -3 Coefficient Matrix

  24. Cramer’s Rule 4x – 2y = -6 -3x + y = -3 Coefficient Matrix

  25. Cramer’s Rule 4x – 2y = -6 -3x + y = -3 Coefficient Matrix

  26. Cramer’s Rule 4x – 2y = -6 -3x + y = -3 Coefficient Matrix

  27. Cramer’s Rule 4x – 2y = -6 -3x + y = -3 Coefficient Matrix

  28. Cramer’s Rule 4x – 2y = -6 -3x + y = -3 Coefficient Matrix

  29. Cramer’s Rule 4x – 2y = -6 -3x + y = -3 Coefficient Matrix

  30. Cramer’s Rule 4x – 2y = -6 -3x + y = -3 ( 6 , 15 ) Coefficient Matrix

  31. Practice Find the determinate of the matrix.

  32. Practice Use Cramer’s Rule to solve the system. (I’ve given the determinate to you already.) x – y = 2 2x + 3y = 14

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