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Understanding Determinants and Cramer’s Rule in Linear Algebra

This lesson presents the concepts of determinants and Cramer's Rule, integral in solving systems of linear equations. Students will learn to find determinants for 2x2 and 3x3 matrices and apply Cramer’s Rule effectively. The lesson includes examples and practice problems that guide students through the processes of determining the values of variables in a system of equations. Additionally, the importance of determinants as measures in linear algebra is emphasized. The lesson includes quizzes to reinforce learning outcomes.

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Understanding Determinants and Cramer’s Rule in Linear Algebra

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  1. 4-4 Determinants and Cramer’s Rule Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

  2. 3x + y = 15 3x – 2y = 6 x + 2y = 18 –2x – 4y = –36 2x + 3y = 35 4x + 6y = 75 Warm Up Solve for x and y. 1. 2. 3. one infinitely many zero

  3. Objectives Find the determinants of 2  2 and 3  3 matrix. Use Cramer’s rule to solve systems of linear equations.

  4. Every square matrix (n by n) has an associated value called its determinant, shown by straight vertical brackets, such as . The determinant is a useful measure, as you will see later in this lesson.

  5. Reading Math The determinant of Matrix A may be denoted as det A or |A|. Don’t confuse the |A| notation with absolute value notation.

  6. Example 1A: Finding the Determinant of a 2 x 2 Matrix Find the determinant of each matrix. Find the difference of the cross products. = 8 – 20 The determinant is –12.

  7. The determinant is. = 1 – You Try Find the determinant of each matrix. .

  8. The determinant D of the coefficient matrix is The coefficient matrix for a system of linear equations in standard form is the matrix formed by the coefficients for the variables in the equations.

  9. Example 2A: Using Cramer’s Rule for Two Equations Use Cramer’s rule to solve each system of equations.

  10. Example 2B: Using Cramer’s Rule for Two Equations Use Cramer’s rule to solve each system of equations.

  11. You Try Use Cramer’s rule to solve. -x + 3y = 7 2x + 3y = -7

  12. To apply Cramer’s rule to 3  3 systems, you need to find the determinant of a 3  3 matrix. One method is shown below.

  13. Example 3: Finding the Determinant of a 3  3 Matrix Find the determinant of M. Step 1 Multiply each “down” diagonal and add. 2(2)(8)+ 4(3)(1) + 1(5)(4) = 64 Step 2 Multiply each “up” diagonal and add. 1(2)(1)+ 4(3)(2) + 8(5)(4) = 186

  14. Example 3 Continued Step 3 Find the difference of the sums. 64 – 186 = –122 The determinant is –122. Check Use a calculator.

  15. You Try Find the determinant .

  16. Lesson Quiz Find the determinant of each matrix. 1. 2. 3. 4. Jeff buys 7 apples and 4 pears for $7.25. At the same prices, Hayley buy 5 apples and 9 pears for $10.40. What is the price of one pear? 6 –34 Use Cramer’s rule to solve. $0.85

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