# Texas Instruments TI-83, TI-83 Plus Graphics Calculator I.1 Systems of Linear Equations

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5 Technology Tip: Now, when you press the variable key X,T,,n you get a T because the calculator is in parametric mode. Also, look at the new WINDOW menu (Figure I.9). In the standard window, the values of T go from 0 to in steps of with the view from 10 to 10 in both directions. But here the viewing rectangle has , been changed to extend from -2 to 2 in both directions. Press GRAPH to see the parametric graph (Figure I.10). I.1.7 Solving Linear Systems: The solutions to a system of equations correspond to the points of intersection of their graph. As an example, let s graph and solve the system 3x y 1 2x y 0. First transform each equation by solving for y: y 3x 1. Then press Y =and enter 3x 1 for Y 1 and y 2x 2x for Y 2 (Figure I.11). 2 Figure I.11: 3x 1 for Y 1 and 2x for Y 2 Find the coordinates of a point of intersection of the two graphs by pressing 2nd CALC 5 [intersect]. Trace with the cursor keys or first along one graph near an intersection and press ; then trace with the cursor along the other graph and press. Marks + are placed on the graphs at these points. Finally, move the cursor near the point of intersection (Figure I.12) and press again. Coordinates of the intersection will be displayed at the bottom of the window. Figure I.12: Setting up to locate the intersection Figure I.13: Point of intersection The TI-83 also has a numerical solver that you can use to solve a linear system. The technique is based on the fact that any solution of the system y 3x 1 is a root of the equation 3x 1 2x, which is equivalent y 2x to the equation 3x 1 2x 0. So press MATH 0 [Solver...] or MATH and enter 3x 1 2x then press ALPHA SOLVE for the x-coordinate of the point of intersection. Then to calculate its y-coordinate, save this value as x (press STO X,T,,n ) and evaluate either 3x 1 or 2x. If this function has already been used on your calculator, press to obtain the solver screen and enter the equation. TI-83, TI-83 Plus Graphics Calculator Copyright by Houghton Mifflin Company. All rights reserved. I-5

7 Figure I.17: Identity matrix and random matrix I.1.9 Row Operations: Here are the keystrokes necessary to perform elementary row operations on a matrix. Your textbook provides more careful explanation of the elementary row operations and their uses. To interchange the second and third rows of the matrix [A] that was defined above, press MATRX ALPHA C MATRX 1, 2, 3 ) (see Figure I.18). The format of this command is rowswap(matrix, row1, row2). To add row 2 and row 3 and store the results in row 3, press MATRX ALPHA D MATRX 1, 2, 3 ) Enter. The format of this command is row+(matrix, row1, row2). Figure I.18: Swap rows 2 and 3 Figure I.19: Add -4 times row 2 to row 3 To multiply row 2 by -4 and store the results in row 2, thereby replacing row 2 with new values, press MATRX ALPHA E (-) 4, MATRX 1, 2 ). The format of this command is *row(scalar, matrix, row). To multiply row 2 by -4 and add the results to row 3, thereby replacing row 3 with new values, press MATRX ALPHA F (-) 4, MATRX 1, 2, 3 ) (see Figure I.19). The format of this command is *row+(scalar, matrix, row1, row2). Technology Tip: It is important to remember that your TI-83 does not store a matrix obtained as the result of any row operations. So when you need to perform several row operations in succession, it is a good idea to store the result of each one in a temporary place. You may wish to use matrix [E] to hold such intermediate results. For example, use elementary row operations to solve this system of linear equations: First enter this augmented matrix as [A] in your TI-83: Next store this matrix in [E] (press MATRX 1 STO MATRX 5 ) so you may keep the original in case you need to recall it. x 2y x 3y 2x 5y 3z 5z TI-83, TI-83 Plus Graphics Calculator Copyright by Houghton Mifflin Company. All rights reserved. I-7

8 Here are the row operations and their associated keystrokes. At each step, the result is stored in [E] and replaces the previous matrix [E]. The solution is shown in Figure I.20. Row Operation Keystrokes row+([e], 1, 2) MATRX ALPHA D MATRX 5, 1, 2 ) STO MATRX 5 *row+(-2, [E], 1, 3) MATRX ALPHA F (-) 2, MATRX 5, 1, 3 ) STO MATRX 5 row+([e], 2, 3) MATRX ALPHA D MATRX 5, 2, 3 ) STO MATRX 5 *row(1/2, [E], 3) MATRX ALPHA E 1 2, MATRX 5, 3 ) STO MATRX 5 Figure I.20: Final matrix after row operations Thus z 2, so y 1 and x 1. I.2 Matrices I.2.1 Matrix Addition: From the home screen, you can perform many calculations with matrices. Add two matrices [A] and [B] by pressing MATRX 1 + MATRX 2. Subtraction is similar. I.2.2 Scalar Multiplication: Calculate the scalar multiplication 2 [A] by pressing 2 MATRX 1. For example, use the matrix [A] shown here (Figure I.21). Figure I.21: Matrix [A] To replace matrix [B] by 2 [A], press 2 MATRX 1 STO MATRX 2, or if you do this immediately after calculating 2 [A], press only STO MATRX 2 (see Figure I.22). Press MATRX to verify that the dimensions of matrix [B] have been changed automatically to reflect this new value. I-8 Copyright by Houghton Mifflin Company. All rights reserved. TI-83, TI-83 Plus Graphics Calculator

9 Figure I.22: Matrix [B] I.2.3 Matrix Multiplication: Set the dimensions of matrix [C] to 2 3 and enter this matrix as [C]: For matrix multiplication of [C] by [A], press MATRX 3 x MATRX Figure I.23: Multiplication of two matrices Technology Tip: If you tried to multiply [A] by [C], your TI-83 would signal an error because the dimensions of the two matrices do not permit multiplication this way. I.2.4 Inverse of a Matrix: Create the square matrix as matrix [D] in your calculator. The symbol of the inverse [D] is [D] -1. Then to calculate this inverse, press MATRX 4 x 1. Figure I.24: Inverse TI-83, TI-83 Plus Graphics Calculator Copyright by Houghton Mifflin Company. All rights reserved. I-9

10 I.2.5 Transpose of a Matrix: The transpose of a matrix [A] is another matrix with the rows and columns interchanged. The symbol for the transpose of [A] is [A] T. To calculate [A] T,press MATRX 1 MATRX 2. Figure I.25: Transpose I.3 Determinants I.3.1 Determinants of a Matrix: Enter the 3 3 matrix as [A]: 1 3 To calculate its determinate, ,press MATRX 5. 1 MATRX 1 ). You should find that A 2. I.4 Additional Topics Figure I.26: Determinant Rectangular-Polar Conversion: The 2nd ANGLE menu provides functions for converting between rectangular and polar coordinate systems. These functions use the current MODE settings, so it is a good idea to check the default angle measure before any conversion. For the following examples, the TI-83 is set to radian measure. Given the rectangular coordinates x, y 4, 3, convert from the rectangular coordinates to polar coordinates r, by pressing 2nd ANGLE 5 4, (-) 3 ) to display the value of r. The value of is displayed after you press 2nd ANGLE 6 4, (-) 3 ). Figure I.27: Coordinate conversions I-10 Copyright by Houghton Mifflin Company. All rights reserved. TI-83, TI-83 Plus Graphics Calculator

11 Suppose r, 3,. To convert from the polar coordinates to rectangular coordinates (x, y), press 2nd ANGLE 7 3, 2nd ) for the x-coordinate; next press 2nd ANGLE 8 3, 2nd ) to display the y-coordinate. I.5 Program: Visualizing Row Operations I.5.1 Entering the Program: The TI-83 is a programmable calculator that can store sequences of commands for later replay. Here s a useful program that demonstrates how elementary matrix row operations used in Gauss-Jordan elimination may be interpreted graphically. Press PRGM to access the programming menu. The TI-83 has space for many programs, each called by a name you give it. Create a new program, so press PRGM 1. For convenience, the cursor is blinking A, indicating that the calculator is set to receive alphabetic characters. Enter a descriptive title of up to eight characters, letters or numerals (but the first character must be a letter). Name this program ROWOPS and press to go to the program editor. Figure I.28: Part of program ROWOPS In the program, each line begins with a colon : supplied automatically by the calculator. Any command you could enter directly in the TI-83 s home screen can be entered as a line in a program. There are also special programming commands. Input the program ROWOPS by pressing the keystrokes given in the following listing. You may interrupt program input at any stage by pressing 2nd QUIT. To return later for more editing, press PGRM, move the cursor down to the program s name, and press. Program Line Keystrokes :Disp PRGM 3 2nd A-LOCK E N T E R :Disp A 2 BY 3 MATRIX PRGM 3 2nd A-LOCK A ALPHA 2 2nd A-LOCK BY ALPHA 3 2nd A-LOCK M A T R I X :Disp A B C PRGM 3 2nd A-LOCK A B C :Disp D E F PRGM 3 2nd A-LOCK D E F :Prompt A, B, C PRGM 2 ALPHA A, ALPHA B, ALPHA C :Prompt D, E, F PRGM 2 ALPHA D, ALPHA E, ALPHA F :A [A](1, 1): B [A](1, 2) ALPHA A STO MATRX 1 ( 1, 1 ) ALPHA : ALPHA B STO MATRX 1 ( 1, 2 ) :C [A](1, 3): D [A](2, 1) ALPHA C STO MATRX 1 ( 1, 3 ) ALPHA : ALPHA D STO MATRX 1 ( 2, 1 ) TI-83, TI-83 Plus Graphics Calculator Copyright by Houghton Mifflin Company. All rights reserved. I-11

12 :E [A](2, 2): F [A](2, 3) ALPHA E STO MATRX 1 ( 2, 2 ) ALPHA : ALPHA F STO MATRX 1 ( 2, 3 ) :ClrHome PRGM 8 :Disp ORIGINAL MATRIX PRGM 3 2nd A-LOCK O R I G I N A L M A T R I X :Pause [A] Frac PRGM 8 MATRX 1 MATH 1 : B -1 (C-AX) Y 2 ALPHA ALPHA B x 1 ( ALPHA C ALPHA A X,T,,n ) ALPHA STO VARS 1 2 : E -1 (F-DX) Y 1 ALPHA ALPHA E x 1 ( ALPHA F ALPHA D X,T,,n ) ALPHA STO VARS 1 1 :ZStandard: Pause: ClrHome ZOOM 6 ALPHA : PRGM 8 ALPHA : PRGM 8 :Disp OBTAIN LEADING PRGM 3 2nd A-LOCK O B T A I N L E A D I N G :Disp 1 IN ROW 1 PRGM 3 2nd ALPHA 1 2nd A-LOCK I N R O W ALPHA 1 ALPHA :*row (A -1, [A], 1) [A] MATRX ALPHA E ALPHA A x 1, MATRX 1, 1 ) STO MATRX 1 :Pause [A] Frac: ClrDraw PRGM 8 MATRX 1 MATH 1 ALPHA : 2nd DRAW 1 : (A/B) (C/A-X) Y2 ALPHA ( ALPHA A ALPHA B ) ( ALPHA C ALPHA A X,T,,n ) ALPHA STO VARS 1 2 :DispGraph: Pause: ClrHome PRGM 4 ALPHA : PRGM 8 ALPHA : PRGM 8 :Disp OBTAIN 0 BELOW PRGM 3 2nd A-LOCK O B T A I N ALPHA 0 2nd A-LOCK B E L O W :Disp LEADING 1 IN PRGM 3 2nd A-LOCK L E A D I N G ALPHA 1 2nd A-LOCK I N :Disp COLUMN 1 PRGM 3 2nd A-LOCK C O L U M N ALPHA 1 ALPHA :*row+ (-D, [A], 1, 2) [A] MATRX ALPHA F (-) ALPHA D, MATRX 1, 1, 2 ) STO MATRX 1 :Pause [A] Frac: ClrDraw PRGM 8 MATRX 1 MATH 1 ALPHA : 2nd DRAW 1 : (E-(BD/A)) -1 (F-(DC/A)) Y1 ALPHA ( ALPHA E ( ALPHA B ALPHA D ALPHA A ) ) x 1 ( ALPHA F ( ALPHA D ALPHA C ALPHA A ) ) ALPHA STO VARS 1 1 :DispGraph: Pause: ClrHome PRGM 4 ALPHA : PRGM 8 ALPHA : PRGM 8 I-12 Copyright by Houghton Mifflin Company. All rights reserved. TI-83, TI-83 Plus Graphics Calculator

13 :[A](2, 2) G MATRX 1 ( 2, 2 ) STO ALPHA G :If G 0 PRGM 1 ALPHA G 2nd TEST 2 0 :Then PRGM 2 :*row (G -1, [A], 2) [A] MATRX ALPHA E ALPHA G x 1, MATRX 1, 2 ) STO MATRX 1 :Disp OBTAIN LEADING PRGM 3 2nd A-LOCK O B T A I N L E A D I N G :Disp 1 IN ROW 2 PRGM 3 2nd ALPHA 1 2nd A-LOCK I N R O W ALPHA 2 ALPHA :Pause [A] Frac PRGM 8 MATRX 1 MATH 1 :ClrDraw: DispGraph: Pause: ClrHome 2nd DRAW 1 ALPHA : PRGM 4 ALPHA : PRGM 8 ALPHA : PRGM 8 :Disp OBTAIN 0 ABOVE PRGM 3 2nd A-LOCK O B T A I N ALPHA 0 2nd A- LOCK A B O V E :Disp LEADING 1 IN PRGM 3 2nd A-LOCK L E A D I N G ALPHA 1 2nd A-LOCK I N :Disp COLUMN 2 PRGM 3 2nd A-LOCK C O L U M N ALPHA 2 ALPHA :[A](1, 2) H MATRX 1 ( 1, 2 ) STO ALPHA H :*row+ (-H, [A], 2, 1) [A] MATRX ALPHA F (-) ALPHA H, MATRX 1, 2, 1 ) STO MATRX 1 :Pause [A] Frac: ClrDraw: FnOff 2 PRGM 8 MATRX 1 MATH 1 ALPHA : 2nd DRAW 1 ALPHA : VARS :[A](1, 3) J MATRX 1 ( 1, 3 ) STO ALPHA J :Vertical J 2nd DRAW 4 ALPHA J :DispGraph: Pause: ClrHome PRGM 4 ALPHA : PRGM 8 ALPHA : PRGM 8 :Disp THE POINT OF PRGM 3 2nd A-LOCK T H E P O I N T O F :Disp INTERSECTION IS PRGM 3 2nd A-LOCK I N T E R S E C T I O N I S :Disp X =, [A](1, 3)Frac, Y =, [A](2, 3)Frac PRGM 3 ALPHA X,T,,n 2nd TEST 1 ALPHA, MATRX 1 ( 1, 3 ) MATH 1, ALPHA ALPHA Y 2nd TEST 1 ALPHA, MATRX 1 ( 2, 3 ) MATH 1 :Stop PRGM ALPHA F TI-83, TI-83 Plus Graphics Calculator Copyright by Houghton Mifflin Company. All rights reserved. I-13

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