Algebra II – SOL Jumpstart AII. 11-12-13-14 Name _________________________ AII.11 Matrices Use the graphing calculator to multiply, to find the determinant or inverse, and to solve systems of equations. Multiplying Matrices: Remember, multiplication of Matrices is NOT commutative! NOTE: The inside dimensions must match or you can’t multiply! 0 6 1 9 19 8 4 10 2 4 8 1 7 6 8 3X2 3X3 Product NOT possible--Undefined These are NOT the same number! 2 3 Example 1. [A] = 0 1 4 0 [B] = 1 2 Find the product of [A] and [B]. On the TI graphing calculator, punch the following buttons to perform matrix multiplication: (2nd) MATRIX 1[A] highlighted EDIT ENTER 2 ENTER x 2 ENTER Then plug in the entries. Now hit 2nd QUIT (2nd) MATRIX EDIT 2[B] highlighted ENTER 2 ENTER x 2 ENTER Then plug in the entries. Then hit 2nd QUIT (2nd) MATRIX 1[A] ENTER times (2nd) MATRIX 2[B] ENTER 5 6 Answer is : 1 2 6 1 15 3 2 0 1 Example 2. Multiply these: Answer is : 2 0 26 3 3 4 0 3 5 3 2 Example 3: Multiply: 6 0 Your calculator should give you an error message of DIM 7 1 MISMATCH. This means the multiplication is not possible. Why is that? The dimensions of each matrix are 3 x 1 and 3 x 1. Because the inner dimensions are not the same, it is NOT possible to multiply. Finding the Inverse of a Matrix: Enter the matrix in your calculator. Use (2nd)MATRIX 1A (or whichever letter is the matrix you just entered) and hit the inverse button X 1 (just below the MATH button) and ENTER . To change your entries from decimals to fractions, select the MATH button and hit 1Frac. NOTE: You will get an error message of SINGULAR MAT if you cannot find the inverse. (This means the determinant is equal to 0.) 2 1 7 14 3 1 Example 1: A = Follow the steps to find the inverse. (Answer is : ) 2 4 1 3 7 14 4 1 Example 2: B = Find the inverse. 7 2 2 1 (Answer is: ) 7 4 AII.12/14 Solving Systems: Using Matrices to solve: First, write a matrix equation. The coefficients can be placed in matrix A and the column matrix of answers can be called matrix B. After plugging both in, just use MATRIX 1A X 1 TIMES MATRIX 2B ENTER Show your set-up matrix equation and the answer step for each of the following. 2 x 5 y 53 Example 1: 6 x 7 y 39 Answer: 2 5 x 53 Set up as: 6 7 y 39 3x 5 y 12 Example 2: 2 x 3 y 8 Answer: 3 5 x 12 Set up as: 2 3 y 8 Put the first Matrix in as A and the answer matrix in as B. Then hit: MATRIX 1A X 1 TIMES MATRIX 2B ENTER . Answer matrix looks like: 4 x 4 4 x 4 9 so y 9 so x = -4, y = 9 0 so y 0 so x = -4, y = 0 Solving Linear Programming problems: Remember, the greatest value of the profit function will occur only at the corner points of the shaded region. Example: The graph of the linear programming model to the left consists of polygon ABCD and its interior. Under these constraints, which is the point where the maximum value of 4x + 3y occurs? Answer: Corner points are at the values of A, B, C, and D. Plug in the x and y values of the points into 4x + 3y to see which gives you the greatest value. In this case it will occur at point B. Solving systems by graphing on the calculator: Solve each equation for y, then input the equation into y= Where the graphs intersect is the x and y solutions to the problem. You can also use the ‘intersect’ feature of the calculator: Examples: Solve the system: x - 2y = 14 and x + 3y = 9 1. The graphing calculator will only accept entries that start with y = , so we need to solve these equations for y =. 1. Enter the first equation into Y1. 2. Enter the second equation into Y2. 3. Hit GRAPH. The graphs appear to intersect OFF the window. We need MORE x-values to the right hand side Oops!! They don't cross in the of the graph. Go to WINDOW. Increase the size of window. Xmax. Hit GRAPH. 4. Use the INTERSECT option to find where the two graphs intersect (the answer). 2nd TRACE (CALC) #5 intersect Move spider close to the intersection. Hit ENTER 3 times. 5. Answer: x = 12 and y = -1 Better! Increase X-max to 20. 2. Solve linear quadratic system: y = x2 - 4x - 2 and y = x - 2 1. Enter the first equation into Y1. 2. Enter the second equation into Y2. 3. Hit GRAPH. 4. Use the INTERSECT option twice to find the two locations where the graphs intersect (the answers). 2nd TRACE (CALC) #5 intersect Move spider close to the intersection. Hit ENTER 3 times. 5. Answer: (5,3) and (0,-2) and A.II Systems of Inequalities-Linear Programming Graphing Linear Inequalities Example 1: Graph Enter 2x + 1 into Y1 Arrow to the far left side of Y1 Hit ENTER until the “shade above” symbol is displayed. Hit ZOOM #6 ZStandard (for a 10x10 window) Graph NOTE: You will have to determine whether to draw a solid line or a dotted line for y = 2x + 1. This problem uses a solid line because of the "less than or equal to" sign. The calculator will display a solid line at all times. (The Application program called “Inequalities” will display solid and dotted lines on the graphs.) Example 2: Solve the system: You will need to isolate the y variable in the second equation so that it can be entered into the calculator with the "shade above" indicated. Solving algebraically, (be careful of the direction of the inequality symbol in this problem.) The answer is the double shaded region on the graph. Example 3: Solve Enter all three equations with appropriate "shade above" or "shade below" symbol. Answer is the darkest shading. Zoom Decimal view.