Congratulations! We have two units left to go. The first topic is partial fraction decomposition. Since it is not a large unit, it will not have its own assessment. We will finish it this week. Next week we will start our last unit, which is limits. The assessment for limits will include material on partial fractions. If you combine two fractions through addition or subtraction, you find the common denominator then add or subtract the adjusted numerators to get your answer. For example, x /3 + (x-1) / 4 = (4x / 12) + 3(x-1) / 12 = (4x + 3x-3) /12 = (7x-3) / 12. You can think of partial fractions as doing the opposite. Instead of being asked to add fractions together, you are being given the final sum of (7x-3) / 12 and asked to find the fractions that were added to get that value, in this case x/3 and (x-1)/4. There is a process to do this. It often involves solving a system of equations with multiple variables. This is where knowing how to find determinants on your calculator and using Cramer's Rule becomes useful, especially when solving systems with many variables. Homework - Due Wednesday - 6/3/20 page 720 - #'s 2, 12, 14, 16, 18, 22, 24, 26, 28, 32, 36 Videos: Setting up the partial fraction decomposition - (determines how many linear and quadratic factors you will need to find). www.youtube.com/watch?v=N1lx9yn3SLI&feature=youtu.be Example one with 2 linear factors www.youtube.com/watch?v=WoVdOcuSI0I&feature=youtu.be Example two with 2 linear factors www.youtube.com/watch?v=RFTAVAWHgN0&feature=youtu.be Example with repeated linear factors www.youtube.com/watch?v=6DdwGw_5dvk&feature=youtu.be Example with linear and quadratic factors www.youtube.com/watch?v=prtx4o1wbaQ&feature=youtu.be Example with repeated quadratic factors www.youtube.com/watch?v=Dupeou-FDnI&feature=youtu.be Another example www.youtube.com/watch?v=04RSkBwVzK0&feature=youtu.be
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Attached is the link to an assessment on system of equations and matrices.
Good Luck! docs.google.com/forms/d/e/1FAIpQLSdM-XUecydJQF1UrdyAGChf_v6gVQFD3msb9hwciEWwyvUq5Q/viewform Attached is a review sheet and answer key for systems of equations and matrices. You should finish the assignment by Wednesday, May 27. The assessment will be on Thursday, May 28. There will be a link on the next post to the assessment. Email me with any questions. Stay healthy!
Now that we know how to calculate a determinant, we can use it to solve various problems. Determinants of matrices can be used to solve systems of equations using Cramer's Rule. We can use determinants to solve various problems such as finding the area of a triangle. Below are the notes and videos demonstrating how to use these techniques as well as the answer key for the homework. Homework assignment due Friday May 22. pages.713 – 14: numbers 30, 34, 38, 40, 42, 48, 52 Videos Cramer's Rule using determinants www.youtube.com/watch?v=ItxF3IjC_uw Example of solving a 2 x 2 system of equations using Cramer's Rule www.youtube.com/watch?v=Z0i_LTUKHVA&feature=youtu.be Example of solving a 3x 3 system of equations using Cramer's Rule. www.youtube.com/watch?v=NhSsdEfJIDo&feature=youtu.be Finding the area of a triangle using determinants www.youtube.com/watch?v=zWI9QeQf2JI
A determinant is a number associated with a square matrix that has certain properties which allows us to 1) determine if a matrix has an inverse (If the determinant is 0, the matrix has no inverse) 2) solve system of equations in matrix form. (using Cramer's Rule). We can find the determinant of any square matrix using Minors and Cofactors. There is a simple formula for finding the determinant of a 2 x 2 matrix. There is a very handy shortcut for finding the determinant of a 3 x 3 matrix called the diagonal method. Most prefer this method much more than using minors and cofactors. You can also use your calculator to find the determinant of a matrix as well. Homework assignment due Wednesday - 5/20/20 Page 713 numbers 2-20 evens Videos Finding the determinant of a 2 x 2 matrix, a 3 x 3 using diagonals, 3 x 3 using minors and cofactors www.youtube.com/watch?v=OI07C1HsOuc Examples Determinant of a 2 x 2 www.youtube.com/watch?v=JTeAPYturls&feature=youtu.be Determinant of a 3 x 3 using diagonals www.youtube.com/watch?v=Ibm0jZrxolo&feature=youtu.be www.youtube.com/watch?v=YsPMbFeoAno&feature=youtu.be Determinant of a 3 x 3 using minors and cofactors www.youtube.com/watch?v=umXgFjqPfsk&feature=youtu.be www.youtube.com/watch?v=sa90Tx9RuSg&feature=youtu.be Finding determinants using the calculator www.youtube.com/watch?v=5gMK5v_Sc64
In the basic mathematical operations of addition/ subtraction, multiplication/division we have identity elements and inverses. In addition and subtraction, the identity element is 0. If you add 0 to any number, you get the same number back again. x + identity = x (0 is the identity element in addition) The identity element in multiplication is 1. If you multiply anything by 1, you get your original element. 5 x identity = 5 (1 is the identity element in multiplication) The same thing is true with matrices. Any square matrix (same number of rows and columns) will have an identity matrix where you have 1's along the diagonal from top left to bottom right and zeros everywhere else. If you multiply a matrix by its identity matrix, you get the original matrix back. See the notes and videos for specific examples. Just like basic mathematical operations, matrices also have inverse matrices. In basic addition, the inverse element is the number you add to get the identity element of 0. Ex: 5 + inverse = 0(identity). So the inverse of 5 would be -5. -2 + inverse = 0(identity). So the inverse of -2 is +2. In multiplication, the inverse element for a number is its reciprocal. 5 x inverse = 1(identity). So the inverse of 5 for multiplication is 1/5. -1/2 x inverse = 1(identity). So the inverse for -1/2 is -2. -(1/2) x(-2) = 1(identity element). Matrices, if they are square matrices (same number of rows and columns), also have inverse matrices. If you multiply a matrix by its inverse, the result is the identity matrix. In math terms, Given a matrix A and its inverse matrix A^-1 then A x A^-1 = I (where I is the identity matrix) We can compute the inverse matrix of a given matrix. There are a few ways to do this. For a 2 x 2 matrix, there is a simple formula that can be used. For any square matrix whether 2 x 2 or larger, you can find the inverse by using Gauss-Jordan elimination on a matrix while simultaneously applying the same steps to the identity matrix. The assignments for this next section will involve finding inverse matrices of a given matrix. Homework due Monday, 5/18/20 pages 697-98: Numbers 4, 6, 8, 10, 12, 16 Videos Finding and identifying identity matrices www.youtube.com/watch?v=hPAS6H6xFa0 Formula for finding the inverse of a 2 x 2 matrix www.youtube.com/watch?v=oXV66LG5xUA Finding inverses using the calculator www.youtube.com/watch?v=_fFj4NbLcTU Finding inverse matrices using an augmented matrix (Gauss - Jordan) for sizes larger than 2 x 2 matrices www.youtube.com/watch?v=KBYvP6YG58g Example finding the inverse of a 3 x 3 matrix using Gauss - Jordan www.youtube.com/watch?v=NXCz9dw5Ce0&feature=youtu.be www.youtube.com/watch?v=6KEmzBwCBzI&feature=youtu.be
Up to this point we have been representing equations as systems of matrices. We have done operations on single rows in a matrix. We can also do operation on entire matrices themselves. The basic operations include addition, subtraction, multiplication. We can do matrix equations to solve for a matrix given several matrices. This next section deals with those operations. For the most part, it involves the basic mathematical operations of addition, subtraction, multiplication and division and knowing when you can use them. The videos give examples and more notes are included in files at the bottom of the page along with your assignment for this section. The problems to be done are on pages 684 -685: #'s 2, 4, 6, 8, 10, 12, 30, 32, 38, 40 This assignment is due Thursday, May 14. Videos: Matrix operations - addition, subtraction, and scalar multiplication www.youtube.com/watch?v=iNty4CSFIpU Examples of addition and subtraction www.youtube.com/watch?v=cTKjm5fgqKM&feature=youtu.be Example of scalar multiplication www.youtube.com/watch?v=kDCka2iqE1Y&feature=youtu.be Combining various operations www.youtube.com/watch?v=SYzQPWMUhV8&feature=youtu.be Matrix multiplication www.youtube.com/watch?v=6Hmzu-WKCjc Examples www.youtube.com/watch?v=RqA4HLtJBBs&feature=youtu.be www.youtube.com/watch?v=l--THwUU73g&feature=youtu.be www.youtube.com/watch?v=zAjUyPe-4mI&feature=youtu.be Using the TI-84 calculator www.youtube.com/watch?v=C7Dc414qmlk
Attached is an example with each step worked out, one at a time. Take a look at the file attached below and see where the numbers are coming from in that example. After you look at this example, look at some of the answer keys to the problems again and then check the videos. If you are still stuck, email me the work from one of the problems assigned and show me the work and steps that you take to the point where you run into difficulty.
In Algebra 2 we learned to solve systems of equations with more than two variables. Generally the technique used was elimination. The goal was to get a system that looked like: X + Y+ Z = 6 X - Y + Z = 2 2X - Y + Z = 3 Into a format where we had a triangular row of zeros such that the system was modified to look like: X + Y + Z = 6 Y + Z = 5 Z = 3 (where the coefficients for the missing variables were 0) This allows us to use the value of Z to solve for Y and then to use Y and Z to solve for X. We can also represent this format in matrix format which allows for a shorthand way to write the system as well as various shorter methods to solve systems of equations using matrices. The first assignments and videos below review the solving of a system of equations using elimination without matrices. The next section than represents a way to solve similar systems of equations in matrix format. 9.3 Several Variables pp.657 – 58: 6, 8, 10, 16, 20 9.3 (cont-d) p. 658: 15, 17, 19, 25, 31 9.3 Dependent p.658: 18, 22, 24, 26, 30 These problems should be completed by Thursday, May 7. System of equations with 3 variables part 1 www.youtube.com/watch?v=wIE8KSpb-E8 System of equations with 3 variables - part 2 www.youtube.com/watch?v=5FvY8XLrqmM Examples solving systems using elimination: www.youtube.com/watch?v=3RbVSvvRyeI&feature=youtu.be www.youtube.com/watch?v=EytTXf8_KYA&feature=youtu.be No solution Case www.youtube.com/watch?v=ryNQsWrUoJw&feature=youtu.be Infinite Solution Case www.youtube.com/watch?v=mThiwW8nYAU&feature=youtu.be The notes and answer keys for these problems are at the very end of this post. Representing Systems using Matrices A matrix (plural matrices) is a format to represent data. A matrix represents data with rows and columns of numbers. Rows are horizontal and columns are vertical. A matrix can be used to represent a system of equations. Example: 2X + 3Y = 25 4X - 7Y = 22 can be represented in matrix form as _____ _____ | 2 3 25 | The elements of a matrix are enclosed in the | 4 -7 22 | [ ] symbols. ____ ____ Note that the rows represents each of the equations. The columns represent the various variables and the constants in the equation. X's are the first column. Y's are the second column. The constants are the 3rd column. In this first lesson, we will discuss syntax and notation of matrices, representing systems of equations using matrices, and reducing matrices/solving systems of equations using the method of Gauss -Jordan elimination / row-reduction. Homework: Text page 673: #'s 2, 4, 6, 16, 18, 20 Due: Monday 5/11/20 Homework: Text page 674: #'s 30, 34, 38, 42 Due: Tuesday 5/12/20. (More practice on matrices.) Videos: Basic notation and description of the order or size of a matrix. www.youtube.com/watch?v=ilFJYjfKYjk&feature=youtu.be Changing an equation into matrix format and then reducing the matrix to row-echelon form using row/reduction or Gauss Jordan elimination. www.youtube.com/watch?v=BWBckWPjfpw Solving a system of equations (2 x 2) using an augmented matrix www.youtube.com/watch?v=V8mb5BFmJO0&feature=youtu.be Solving a system of equations with no solutions using matrices www.youtube.com/watch?v=NgXXKmQHFDg&feature=youtu.be Solving a system of equations with infinite solutions using matrices www.youtube.com/watch?v=9_wGz6zcZ6s&feature=youtu.be Solving a system of three equations using a system of matrices www.youtube.com/watch?v=WiVeiVIu_SM&feature=youtu.be Solving a system of equations using matrices and REDUCED row-echelon form. www.youtube.com/watch?v=-bPPDq0Y8s4 Using the TI-84 matrix menu to enter a Matrix and show row operations on a Matrix www.youtube.com/watch?v=ABY5rYu3Rmc&feature=youtu.be Using the TI-84 to show Matrix operations on the home screen of the calculator www.youtube.com/watch?v=syx9LY7qMgo&feature=youtu.be There are more videos at the page www.mathispower4u.com/alg-2.php if you wish to see more examples. On the page, press the control key and the f key to open up a search box and type matrix to see the list of videos. lesson_plan_-_section_9.4_-_matrices_-_day_1.pdf Download File answer_key_-_section_9.4_-_matrices_day_1.pdf Download File answer_key_-_section_9.4_matrices_day_2.pdf Download File
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