Cij = aik ⋅ bkj , for i = 1,2, , m j = 1,2, , n | HP 49g specification

Matrix multiplication

Matrix multiplication is defined by Cm⋅ n = Am⋅ p⋅Bp⋅ n. Notice that matrix multiplication is only possible if the number of columns in the first operand is equal to the number of rows of the second operand. The general term in the product, cij, is defined as

p

cij = aik ⋅ bkj , for i = 1,2, , m; j = 1,2, , n.

∑k =1

Matrix multiplication is not commutative, i.e., in general, A⋅B ≠ B⋅A. Furthermore, one of the multiplications may not even exist. The following screen shots show the results of multiplications of the matrices that we stored earlier:

Term-by-term multiplication

Term-by-term multiplication of two matrices of the same dimensions is possible through the use of function HADAMARD. The result is, of course, another matrix of the same dimensions. This function is available through Function

catalog (			), or through the MATRICES/OPERATIONS sub-menu
(		). Applications of function HADAMARD are presented next:

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HP 49g manual Cij = aik ⋅ bkj , for i = 1,2, , m j = 1,2, , n

Contents

User’s manual Graphing calculatorPage Preface Page Table of Contents Calculations with real numbers Calculations with complex numbers Algebraic and arithmetic operations Solution to equations Operations with lists Vectors Matrices and linear algebra Multi-variate Calculus Applications GraphicsCalculus Applications Vector Analysis Applications Statistical Applications Differential EquationsProbability Distributions Numbers in Different Bases Limited Warranty W-1Service, W-2 Regulatory information, W-4 Batteries Chapter Getting startedBasic Operations Contents of the calculator’s display Associated with the six soft menu keys, F1 through F6 MenusTool menu Setting time and date Introducing the calculator’s keyboardPage ~„p Selecting calculator modesAlpha function, to enter the upper-case letter P Operating Mode 23.0 √ 3.*5.-1/3.*3./23.3+EXP2.5 Lets try now the expression proposed earlier Number Format and decimal dot or comma Page Page Page Angle Measure Coordinate System Selecting CAS settings Explanation of CAS settings Selecting Display modes Selecting the display font Selecting properties of the line editor Selecting properties of the Stack References Selecting properties of the equation writer EQW Editing expressions in the stack Chapter Introducing the calculatorCalculator objects Creating arithmetic expressions Result³5*„Ü1+1/7will be shown„ÜR3as follows -2Q3`.5 Resulting…in³5*„Ü1+1/7the output„ÜR3-2Q3`.5 Creating algebraic expressions + y Using the Equation Writer EQW to create expressions Edited expression*„Ü5+1/3looks as follows + 2 ⋅ 5 + ~‚n Subdirectories Organizing data in the calculatorHome directory For this example, the Home Typing variable names Locks the alphabetic keyboard in lower case. When lockedVariables Name Contents Type Creating variables 3V5K~a12` ³~„r/„Ü Checking variables contents ‚å‚é~„r³„ì Using the right-shift key followed by soft menu key labels Deleting variables Using function Purge in the stack in Algebraic modeUsing function Purge in the stack in RPN mode @PURGE@J@@p1@@` We use Show Prog menu list and select Memory ‚¯ Undo and CMD functionsChoose boxes vs. Soft Menu Page Display now shows Not shown in this screen. To Find it we use To activate the Order command we press the soft menu key Examples of real number calculations Operation key, namely Examples in ALG modeChapter Calculations with real numbers For both, ALG and RPN modes, using the Equation Writer Directly on the stack RPN mode, enter the number first, then the function, e.g Combination When calculating in the stack in ALG modeKey. When „¸Page RPN mode \„¸45`‚¹ Real number functions in the MTH menu Hyperbolic functions and their inverses Pressing shows the remaining options Key or With unitsUnits menu Page Labels will be listed Available unitsFor Set of units Key or Keystroke sequence to navigate Through the menus Attaching units to numbers With the keyboard, e.g.,will produce the entry Using Ubase to convert to the default unit 1 m results Operations with units Different terms in parentheses, e.g 12m 3250mi Unit conversions @@OK@@-˜Physical constants in the calculator These operations produce the following output Copies value with or without units to the stack When selected, constants are shown with units attachedWhen selected, constants are shown without units Exit constants library Which produces Defining and using functionsLike this Display shows what is called a tagged value Here ‚@@@H@@ ‘LNx+1 + EXPx’ Entered by using You could try Chapter Calculations with complex numbers Setting the calculator to Complex modeTo work with complex numbers select the CAS complex mode Definitions To enter the unit imaginary number type the I key Entering complex numbersPolar representation of a complex number Simple operations with complex numbers Cmplx menus Cmplx menu through the MTH menu Cmplx menu in the keyboard Functions applied to complex numbers Function Droite equation of a straight line Function Droite is found in the command catalog Reference Chapter Algebraic and arithmetic operations Entering algebraic objects Simple operations with algebraic objects Press To recover Variable menu @HELP@` Functions in the ALG menu To execute the command Catalog To complete the operation Trig menu, triggered by using Expansion and factoring using log-exp functionsExpansion and factoring using trigonometric functions Operations with transcendental functions Numbers or to polynomials. The remaining options 1. Integer Functions in the Arithmetic menuShows the following menu X6-1 = X5-5*X4+25X3-125X2+625X-3125X+5+15624 PolynomialsHorner function Proot function VariablePcoef function Quotient and Remainder functions SIMP2 function FractionsPeval function Fcoef function Propfrac functionPartfrac function Step-by-step operations with polynomials and fractions Froots function Reference Function Isol Chapter Solution to equationsSymbolic solution of algebraic equations Function Solve Page Function Solvevx Function Zeros Numerical solver menu Solve equation Polynomial EquationsEnter vector of coefficients Page Page Financial calculations Solving equations with one unknown through NUM.SLV Solution to simultaneous equations with Mslv Solution shown is X 4.5006E-2Page Reference Changing sign Chapter Operations with listsCreating and storing lists Addition, subtraction, multiplication, division Example When applied to lists, acts a concatenation operatorFunctions‚Napplied to lists MTH/LIST menu Lists of complex numbersLists of algebraic objects Inverse 1/x Calculate product of elements in the list Calculate increment among consecutive elements in listCalculate summation of elements in the list Sorts elements in increasing order MAP function Reference SEQ function, available through the command catalog , takesSEQ function Typing vectors in the stack Chapter VectorsEntering vectors General row vector Vector Vector of algebraics Storing vectors into variables in the stack Mode Then, ‚íinVectors can also be entered by using the matrix writer Stored into variables Key is used to edit the contents of a selected cell WID Function Delete Will delete Key will add a column full of zeros at the locationCell of the spreadsheet Selected Used in the following exercises Simple operations with vectorsAddition, subtraction Multiplication by a scalar, and division by a scalar Absolute value function Dot product MTH/VECTOR menuMagnitude Examples Cross product Chapter Matrices and linear algebra Entering matrices in the stack Display in RPN mode will look very similar to these Typing in the matrix directly into the stack Operations with matrices Addition and subtraction Multiplication RPN mode, try the following eight examples Cij = aik ⋅ bkj , for i = 1,2, , m j = 1,2, , n Page Function DET Solution of linear systemsCharacterizing a matrix The matrix Norm menu Function Trace 2x1 + 3x2 -5x3= Using the numerical solver for linear systemsPress Following input form will be provide right Can be written as the matrix equation A⋅x = b, if @SOLVE Solution with the inverse matrix Solution by division of matrices Chapter Graphics Graphs options in the calculator As an example, lets plot the function Them simultaneously if in RPN mode Use a range of -4 toPlotting an expression of the form y = fx Here is picture To recover the menu, and return to the PlotAlso, check that for x = -1.48 , y = Press Then To see the table, press Soft menu key You will be returned to normal calculator displayZoom factor = 0.5. Toggle Suggested early, namely x = Option Decimal Produces x-increments Option Integer Fast 3D plotsKey simply changes the font in the table from small to big, To return to normal calculator display press You can change Orientation Find Fast3DValues of 10 and 8 for the Step data @EDITL Reference$ Press To leave the Edit environment Limits and derivatives Chapter Calculus ApplicationsCalc Calculus menu Function lim Menu within the Calcl menu Infinity symbol is associated with Key, .eIncluding some limits If and only if, fx = dF/dx, and C = constant Anti-derivatives and integralsFunctions INT, INTVX, RISCH, Sigma and Sigmavx Xdx = F b − F a Functions TAYLR, TAYLR0, and Series ⋅ x − x o n Reference Chapter Multi-variate Calculus Applications Partial derivatives Multiple integrals Y dA = b Φ x , y dydx = Gradient Chapter Vector Analysis ApplicationsDel operator Alternatively, use function Deriv as follows DivergenceCurl CALC/DIFF menu Solution to linear and non-linear equationsChapter Differential Equations Ldec Function Ldec = K1⋅e-3x+ K2⋅e5x + K3⋅e2x + 450⋅x2+330⋅x+241/13500 Function DesolveWhich can be simplified to Variable Odetype DESOLVE` Laplace Transforms Laplace transform and inverses in the calculator Where Function FOURIER`Fourier series for a quadratic function Exercises. Function Collect is available in the ALG menu Thus = 1/3, c Factorials, combinations, and permutations Chapter Probability DistributionsMTH/PROBABILITY.. sub-menu part Random numbers Student-t distribution MTH/PROB menu part„´The Normal distribution Chi-square distribution F distribution Chapter Statistical Applications Entering data‚Ù5 Sample vs. population Now Obtaining frequency distributions Can be used Fitting data to a function y = fxƑ3.Fitdata Lines in RPN mode Using Key select all the options for outputs, i.e When that field is selected StatisticsTo access the summary stats… option, use Obtaining additional summary statistics Offers the following options Interpreted as follows Confidence intervalsPress to obtain the following results @GRAPH Interval information Corresponding interval limits 21.88424 and 24.51576. Press Return to the previous results screen, and/or press @@@OK@@@ Select ∝ ≠ 150. Then, press Result is Writing non-decimal numbers Chapter Numbers in Different BasesBase menu With system flag 117 set to Soft menus, the Base menu shows @HEX Storing objects in the SD card Using SD cardsChapter Recalling an object from the SD card Purging an object from the SD card Limited Warranty Service Canada 905 206-4663 or 800- HP Rotc = Rest of the country Regulatory information USA