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Chapter 4

Chapter 4. Section 6. Determinants and Cramer’s Rule. Evaluate 2 x 2 determinants. Use expansion by minors to evaluate 3 x 3 determinants. Understand the derivation of Cramer’s Rule. Apply Cramer’s Rule to solve linear systems. 1. 2. 3. 4. 4.6. 1. Objective.

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Chapter 4

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  1. Chapter 4 • Section 6

  2. Determinants and Cramer’s Rule Evaluate 2 x 2 determinants. Use expansion by minors to evaluate 3 x 3 determinants. Understand the derivation of Cramer’s Rule. Apply Cramer’s Rule to solve linear systems. 1 2 3 4 4.6

  3. 1 Objective • Evaluate 2 x 2 determinants.

  4. A square matrix has the same number of rows as columns. Associated with every square matrix is a real number called a determinant. Square matrix: Determinant:

  5. There is a formula we can use to evaluate a 2 x 2 square matrix: Value of a 2 x 2 determinant

  6. EXAMPLE 1 Evaluate the determinant.

  7. continued Here and so

  8. 2 Objective • Use expansion by minors to evaluate 3 x 3 determinants.

  9. To calculate a 3 x 3 determinant , we rearrange terms using the distributive property. Each quantity in parentheses represents a 2 x 2 determinant that is the part of the 3 x 3 determinant that remains when the row and column of the multiplier are eliminated, as shown next.

  10. Eliminate the first row and first column. Eliminate the second row and first column. Eliminate the third row and first column.

  11. The minors of the elements and are, respectively, , and, .

  12. Expansion of a Determinant by Minors Multiply each element in the first column by its minor and combine those products.

  13. EXAMPLE 2 Evaluate the determinant using expansion by minors about the first column.

  14. continued In this determinant and . Multiply each of these numbers by its minor, and combine the three terms using the definition. Notice that the second term in the definition is subtracted.

  15. continued

  16. continued To get a 0 for the third element in the first row, multiply row 3 by – 1 and add to the first row. To get a 0 for the third element in the second row, multiply the elements of the third row by .

  17. continued This last operation shows that the inverse is the matrix,

  18. 3 Objective • Use inverse matrices to solve systems of linear equations.

  19. Given the linear system The definition of matrix multiplication can be used to rewrite the system as If , and

  20. Then the system becomes . If exists, then we can solve the system by multiplying by it as follows Matrix gives the solution to the system.

  21. EXAMPLE 3 Use the inverse of the coefficient matrix to solve the system.

  22. continued For this system , , and . We first need to find the inverse of A. Now we find the product . The solution to the system is x = 3, y = 5.

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