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Linear System of Simultaneous Equations Warm UP

Linear System of Simultaneous Equations Warm UP. First precinct: 6 arrests last week equally divided between felonies and misdemeanors. Second precinct: 9 arrests - there were twice as many felonies as the first precinct.

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Linear System of Simultaneous Equations Warm UP

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  1. Linear System of Simultaneous Equations Warm UP First precinct: 6 arrests last week equally divided between felonies and misdemeanors. Second precinct: 9 arrests - there were twice as many felonies as the first precinct. Write a system of two equations and find out how many felonies and misdemeanors occurred.

  2. Sections 4.1 & 4.2 Matrix Properties and Operations

  3. Algebra

  4. Matrix A matrix is any doubly subscripted array of elements arranged in rows and columns enclosed by brackets. Element

  5. Dimensions of a Matrix 3, 2

  6. Name the Dimensions

  7. Row Matrix [1 x n] matrix

  8. Column Matrix [m x 1] matrix

  9. Square Matrix Same number of rows and columns Matrices of nth order -B is a 3rd order matrix

  10. The Identity

  11. Identity Matrix Square matrix with ones on the diagonal and zeros elsewhere.

  12. The Equal

  13. Equal Matrices Two matrices are considered equal if they have the same dimensions and if each element of one matrix is equal to the corresponding element of the other matrix. = Can be used to find values when elements of an equal matrices are algebraic expressions

  14. To solve an equation with matrices • 1. Write equations from matrix • 2. Solve system of equations • Examples • = • =

  15. Linear System of Simultaneous Equations How can we convert this to a matrix?

  16. Matrix Addition A new matrix C may be defined as the additive combination of matrices A and B where: C = A + B  is defined by: Note: all three matrices are of the same dimension

  17. Addition If and then

  18. Matrix Subtraction C = A - B Is defined by

  19. Subtraction If and then

  20. Matrix Addition Example

  21. Multiplying Matrices by Scalars

  22. Matrix Operations

  23. Matrix Multiplication Matrices A and B have these dimensions: Video [r x c] and [s x d]

  24. Matrix Multiplication Matrices A and B can be multiplied if: [r x c] and [s x d] c = s

  25. Matrix Multiplication The resulting matrix will have the dimensions: [r x c] and [s x d] r x d

  26. A x B = C [2 x 2] [2 x 3] [2 x 3]

  27. A x B = C [3 x 2] [2 x 3] Result is 3 x 3 [3 x 2] [2 x 3] A and B can be multiplied [3 x 3]

  28. Practice

  29. Combing Steps

  30. Matrices In the Calculator • 2nd x-1 button • must enter dimensions before data • must enter the matrix before doing calculations

  31. 2.5 Determinants of 2 X 2 Matrix

  32. Example • Find the value of • = 3(9) - 2(5) or 17

  33. Determinants of 3 X 3 Matrix Third-order determinants - Determinants of 3 × 3 matrices are called  Expansion by diagonals Step 1: begin by writing the first two columns on the right side of the determinant, as shown below Step 2: draw diagonals from each element of the top row of the determinant Downward to the right.  Find the product of the elements on each diagonal.

  34. Determinants of 3 X 3 Matrix Step 3: draw diagonals from the elements in the third row of the determinant upward to the right.  Find the product of the elements on each diagonal. Step 4: To find the value of the determinant, add the products of the first set of diagonals and then subtract the products of the second set of diagonals.  The value is:

  35. Example

  36. Determinants of 3 X 3 Matrix expansion by minors.  The minor of an element is the determinant formed when the row and column containing that element are deleted.  

  37. Example . = 2 -(-3) +(-5) = 2(-8) + 3(-11) – 5(-7) = -14

  38. Inversion

  39. Matrix Inversion Like a reciprocal in scalar math Like the number one in scalar math

  40. Inverses

  41. 2.5 Inverses Step 1 : Find the determinant. Step 2 : Swap the elements of the leading diagonal. Recall: The leading diagonal is from top left to bottom right of the matrix. Step 3: Change the signs of the elements of the other diagonal. Step 4: Divide each element by the determinant.

  42. Example . First find the determinant = 4(2) - 3(2) or 2 or .

  43. Solving Systems with Matrices Step 1: Write system as matrices Step 2: Find inverse of the coefficient matrix. Step 3: Multiply each side of the matrix equation by the inverse Constant Matrix Coefficient Matrix Variable Matrix

  44. Example • Solve the system of equations by using matrix equations. 3x + 2y = 3 2x – 4y = 2

  45. Example Write the equations in the form ax + by = c 2x – 2y – 3 = 0 ⇒ 2x – 2y = 3 8y = 7x + 2 ⇒ 7x – 8y = –2 Step 2: Write the equations in matrix form. Step 3: Find the inverse of the 2 × 2 matrix. Determinant = (2 × –8) – (–2 × 7) = – 2       Step 4: Multiply both sides of the matrix equations with the inverse

  46. Using the calculator • How do you use the calculator to find the solution to a system of equations? • Put both coefficient and answer matrix into calculator • Multiply the inverse of the coefficient matrix and the answer matrix to get values. 3x + 2y = 3 x + 2y+3z=5 2x – 4y = 2 3x+2y-2z=-13 5x+3y-z=-11

  47. Modeling Motion with Matrices • Vertex Matrix – A matrix used to represent the coordinates of the vertices of a polygon • Transformations -Functions that map points of a pre-image onto its image • Preimage-image before any changes • Image-image after changes • Isometry-a transformation in which the image and preimage are congruent figures • Translation – a figure moved from one location to another without cahnging sizze, shape, or orientation

  48. Translation • Suppose triangle ABC with vertices A(-3, 1), B(1, 4), and C(-1, -2) is translated 2 units right and 3 units down. •  a. Represent the vertices of the triangle as a matrix. • b. Write the translation matrix. • c. Use the translation matrix to find the vertices of A’B’C’, the translated image of the triangle. • d. Graph triangle ABC and its image.

  49. Translation • a. The matrix representing the coordinates of the vertices of triangle ABC will be a 2  3 matrix. • b. The translation matrix is . • c. Add the two matrices. • d. Graph the points represented by the resulting matrix.

  50. Example ΔX'Y'Z' is the result of a translation of ΔXYZ. A table of the translations is shown. Find the coordinates of Y and Z'. Solve the Test Item • Write a matrix equation. Let (a, b) represent the coordinates of Y and let (c, d) represent the coordinates of Z'. • Since these two matrices are equal, corresponding elements are equal. Solve an equation for x. Solve an equation for y. –3 + x = 4 2 + y = 7 x = 7 y = 5 • Use the values for x and y to find the values for Y(a, b) and Z' (c, d). a = –4 b = –5 9 = c 4 = d

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