hp calculators hp 39g+ & hp 39g/40g Using Matrices How are matrices stored? How do I solve a system of equations? Quick and easy roots of a polynomial

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1 hp calculators hp 39g+ Using Matrices Using Matrices The purpose of this section of the tutorial is to cover the essentials of matrix manipulation, particularly in solving simultaneous equations. How are matrices stored? How do I solve a system of equations? Quick and easy roots of a polynomial Geometric transformations

2 How are matrices stored? Matrices are stored and edited in the Matrix Catalog. It can hold matrices that are vectors or contain complex numbers. Once defined, a matrix or vector can be used or manipulated in any other view. Some extremely powerful matrix function are included in the MATH menu (see The MATH menu). It can be cleared using SHIFT CLEAR. 2 3 Given that A = 1 4 and 0 4 B = 2 3, 1 find the value of B A Enter the Matrix Catalog by pressing SHIFT MATRIX. With the highlight on M1, press EDIT. Enter the values 2 and 3, pressing ENTER after each. Press down arrow to begin a new line and enter the values 1 and 4. Leave the edit view by pressing SHIFT MATRIX again. Use the same method to create M2. Now in the HOME view, perform the calculation, storing the solution in M3 in case it is required later. The value displayed can be viewed more easily using SHOW. Key sequence SHIFT MATRIX EDIT 2 ENTER 3 ENTER (-) 1 ENTER 4 ENTER SHIFT MATRIX EDIT 0 ENTER 4 ENTER 2 ENTER 3 ENTER HOME ALPHA M 2 SHIFT x -1 * ALPHA M 1 STO> ALPHA M 3 ENTER SHOW

3 How do I solve a system of equations? I Systems of simultaneous linear equations of any size can be solved either with an inverse matrix or using the function RREF from the MATH menu. The sales of type A, B and C computers for three successive weeks are shown right. Find the prices of each type of computer. A B C Value Week $12,900 Week $13,335 Week $13, A This system of equations can be represented in matrix form as: B = C If we enter the first matrix as M1 and the values as M3 then the solution is M1-1 *M3. Change to the Matrix Catalog and enter the M1 as outlined on the previous page. Before entering the values into M2, press to change it to. This moves the cursor down instead of right after each entry. Rounding error may make it difficult to see the solution in the HOME view. Viewing M3 is probably the better option. Key sequence SHIFT MATRIX EDIT 2 ENTER 0 ENTER 4 ENTER 3 ENTER 5 ENTER 0 ENTER 1 ENTER 4 ENTER 2 ENTER SHIFT MATRIX EDIT ENTER ENTER ENTER HOME ALPHA M 1 SHIFT x -1 * ALPHA M 2 STO> ALPHA M 3 ENTER SHIFT MATRIX EDIT

4 How do I solve a system of equations? II If there is no valid solution then the matrix inverse will not exist, as shown right. Here it is better to use the RREF function (Reduced Row Echelon Form). It acts on an augmented matrix and its advantage is that it will work even if the equations are inconsistent. Three cases can result, as below. Case 1: A unique solution 2x+ 4z = x+ 5y = 4 becomes x+ 4y+ 2z = The final column of the matrix contains the solution. Case 2: No solution 2x+ y z = 2 3x+ 5y = 1 becomes x+ 4y+ z = In this case the final line of the reduced row echelon matrix is [ ], hence no solution. Case 3: Infinite solutions This case is similar to that of no solution but the final line of the reduced row echelon matrix will be [ ], which corresponds to the case of infinite solutions.

5 Quick and easy roots of a polynomial I Although it is possible to find roots of any graph by using the PLOT view and FCN, the MATH menu provides a function called POLYROOT which will find all the roots of any polynomial in one operation. Coefficients must be supplied in the form of a row vector using square brackets and solutions are returned the same form. If any of the roots are complex then the entire set will be returned in complex form (a,b) as in the example below. Find the roots of ( ) 4 2 f x = x 27x 14x Note: A worthwhile tip here is to store the solution vector into a matrix variable. This allows easy viewing of the solutions, both real and complex and is the method used in the keystroke list below. Key sequence MATH 7 ENTER SHIFT [ 1, 0, (-) 27, (-) 14, 135 SHIFT ] ) STO> ALPHA M 1 ENTER SHIFT MATRIX EDIT HOME ALPHA M 1 ( 1 )

6 Quick and easy roots of a polynomial II This method also allows the solutions to be easily accessed in the HOME view as shown right, or in any other view. The roots, which are the individual elements of the matrix, can be referenced as M1(1), M1(2) etc or whichever matrix you used. To access only the real component you could use the RE command, for example as RE(M1(1)), or just edit the value to remove the brackets.

7 Geometric transformations I 1 y Geometric transformations can be done using matrices and affine equations. For example, suppose we define a letter E as shown on the right. We will begin by having the calculator 1 display this by storing the vertices of the letter in matrix M1. This matrix is shown below as a composite image. The bottom row of 1 s are for use with the affine transformation matrices. x Begin by entering the equations shown right into the SYMB view of the Parametric aplet. Set the axes as shown, particularly the values of TRng and TStep which are chosen to walk through the elements of matrix M1. The result is the PLOT view shown. Notice that as T takes the values 1, 2,. 13, the (X1,Y1) coordinates plotted are those stored in the matrix. Using a shape with more or with less vertices would require changing TRng but not TStep.

8 Geometric transformations II Suppose that we want to reflect this letter in the y axis and then translate it a distance of 5. The transformations to do this are: 2 x' 1 0 x 0 Reflection: = + y ' 0 1 y 0. This can be expressed as an affine matrix as: x ' x y' = y x'' 1 0 x' 5 Similarly the translation: = + can be expressed y'' 0 1 y' 2 x '' x ' as the affine transformation given by y'' = y' The two transformations can be combined into one and stored in M3 by performing the first of the calculations shown right. This matrix is now used to transform the letter E by performing the second of the calculations. The resulting matrix of transformed vertices is stored in M2 for display in the same way as the original version.