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20010102

Rref

Function: Returns the reduced row echelon form of a matrix.

Syntax: Rref Mat

○ ○ ○ ○ ○

Example

To determine the reduced row echelon form of the matrix below

– 2

– 6

– 1

– 9

– 5

– 4

(MAT)

(CALC)

(Rref)

( [ )

c,a,

( ] )

( [ )

b,j,

( ] )

( [ )

f,c,e,

( ] )

Ref

Function: Returns the row echelon form of a matrix.

Syntax: Ref Mat

○ ○ ○ ○ ○

Example

To determine the row echelon form of the matrix below

– 2

– 6

– 1

– 9

– 5

– 4

(MAT)

(CALC)

(Ref)

( [ )

c,a,

( ] )

( [ )

b,j,

( ] )

( [ )

f,c,e,

( ] )

7-1-33

Using the CAS (Computer Algebra System) Mode

147

–

1 0 0

0 1 0

0 0 1

–

1 1 0

0 1

0 0 1

–

Содержание ALGEBRA FX 2.0 Plus

Страница 1: ...10102 Computer Algebra System and Tutorial Modes ALGEBRA FX 2 0 PLUS only 7 1 Using the CAS Computer Algebra System Mode 7 2 Algebra Mode 7 3 Tutorial Mode 7 4 Algebra System Precautions Chapter 7 20010102 ...

Страница 2: ...gebra Mode is performed in the upper part of the display which is called the input area You can input commands and expressions at the current cursor location Calculation results appear in the lower part of the display which is called the output area When a calculation produces an equation or inequality the lower part of the display is divided between a natural result display area for the result an...

Страница 3: ... Data Matrix m n 1 1 entry 1 2 entry 1 m entry 2 1 entry 2 n entry m n entry m n entry The above input is arranged to show the relative positions of entries in the matrix Actual input is an unbroken line from left to right Entries should be separated by commas and the entire set of elements should be enclosed within square brackets And each line also should be enclosed within square brackets You c...

Страница 4: ...ode Function menu command input Manual formula and parameter input k k k k k Menu Command Input Press a function menu key to display the menu of functions for the type of operation you are trying to perform TRNS formula transformation menu CALC formula calculation menu EQUA equation inequality menu eqn calls up an equation stored in Equation Memory in accordance with a specified input value CLR va...

Страница 5: ... sinh cosh tanh sinh 1 cosh 1 tanh 1 Jkey Yn rn Xtn Ytn Xn input of graph memory Yn rn Xtn Ytn Xn k Formula Memory The CAS Mode has 28 formula variables Variable names are the letters A through Z plus r and θ CAS Mode formula variables are independent of standard value variables Example To assign a formula that differentiates sin X at X cos X to variable A 2 CALC b diff sv v aav A w 7 1 4 Using th...

Страница 6: ... assigned to it ah M aav A b c w Example To recall the value of variable A when the list X Y Z is assigned to it av A w Example To recall the first component A 1 of variable A when vector XY Z is assigned to it av A b w 7 1 5 Using the CAS Computer Algebra System Mode 1 2 3 X Y Z ...

Страница 7: ...h memory Y1 at X 2 CALC b diff J1 Yn b v w k Eqn Memory When a calculation result is an equation or inequality its formula number is displayed in the formula number area and the equation is stored in Eqn memory 1 Stored equations can be recalled with the eqn command rclEqn command or rclAllEqn command 7 1 6 Using the CAS Computer Algebra System Mode 1 Up to 99 formulas can be stored in Eqn memory ...

Страница 8: ...ion performed After a calculation or after pressing A you can press f or c to recall previous formulas k Moving the Cursor Between Display Areas When indicates a calculation result that does not fit on the display the cursor keys perform output area scrolling To use the Replay Function from this condition press 6 g 2 SW change to a dotted line display to indicate that cursor key operations control...

Страница 9: ... and 3 TYPE functions while the graph formula screen is on the display Press 6 DRAW to draw a graph k RECALL ANS Function Pressing 6 g 3 R ANS recalls Ans Memory contents k Solution Memory In the CAS Mode or ALGEBRA Mode you can save the history of a calculation you perform replay memory contents into solution memory This section describes how you can access and work with the contents of solution ...

Страница 10: ... 1 YES returns to the solution memory initial screen without saving anything u To clear solution memory contents Clear Memory On the initial solution memory screen press 2 DEL A Press 1 YES to clear solution memory contents Pressing i returns to the solution memory initial screen Pressing 6 NO in place of 1 YES returns to the solution memory initial screen without clearing anything This clears bot...

Страница 11: ... w Yes to delete the record you displayed To clear the above screen without deleting anything press i No To toggle record number display on and off Press 4 NUM to toggle display of the record number on and off u To optimize solution memory Optimization On the initial solution memory screen press 3 OPT Pressing i returns to the solution memory initial screen Optimizing solution memory rearranges da...

Страница 12: ...hin square brackets can be omitted u expand Function Expands an expression Syntax expand Exp Eq Ineq List Mat Vect Example To expand X 2 2 1 TRNS b expand v c xw X2 4X 4 u rFactor rFctor Function Factors an expression up to its root Syntax rFactor Exp Eq Ineq List Mat Vect Example To factor the X2 3 1 TRNS c rFctor vx dw X 3 X 3 u factor Function Factors an expression Syntax factor Exp Eq Ineq Lis...

Страница 13: ...s the addition theorem to expand a trigonometric function Syntax tExpand Exp List Mat Vect Example To employ the addition theorem to expand sin A B 1 TRNS f TRIG b tExpnd s av A al B w cos B sin A sin B cos A u tCollect tCollc Function Employs the addition theorem to transform the product of a trigonometric function to a sum Syntax tCollect Exp List Mat Vect Example To employ the addition theorem ...

Страница 14: ...G d trigToE c a i vw u expToTrig expToT Function Converts an exponential function to a trigonometric or hyperbolic function Syntax expToTrig Exp List Mat Vect Example To convert eix to a trigonometric function 1 TRNS f TRIG e expToT I ex a i vw cos X sin X i u simplify smplfy Function Simplifies an expression Syntax simplify Exp Eq Ineq List Mat Vect Example To simplify 2X 3Y X 3 Y X 3Y 3 X 1 TRNS...

Страница 15: ...ect Exp Eq Ineq List Mat Vect Exp variable Example To rearrange X2 AX BX focusing on the variable X 1 TRNS i collct vx av A v al B vw X2 A B X X is the default when nothing is specified for Exp variable u substitute sbstit Function Assigns an expression to a variable Syntax substitute Exp Eq Ineq List Mat Vect variable expression variable expression Example To assign 5 to X in 2X 1 1 TRNS j sbstit...

Страница 16: ...xample 920 Normal jMcaw 12157665459056928801 approx 1 TRNS l approx jMcaw 1 215766546E 19 Display Norm1 7 1 15 Using the CAS Computer Algebra System Mode About approx With normal calculations when approx is not used in the CAS Mode calculation results are displayed in full without using exponents When you use approx in the CAS Mode however results are displayed using the exponential format range s...

Страница 17: ... Integrates an expression Syntax Exp List variable integration constant Exp List variable integration constant Exp List variable lower limit upper limit Example To integrate X2 with respect to X 2 CALC c vxw X is the default when no variable is specified u lim Function Determines the limits of a function expression Syntax lim Exp List variable point direction Example To determine the limits of sin...

Страница 18: ...s from X 1 through X 5 2 CALC f Π vx v b fw 14400 u taylor Function Finds a Taylor polynomial Syntax taylor Exp List variable order center point Example To find a 5th order Taylor polynomial for sin X with respect to X 0 2 CALC g taylor sv v f aw The default center point is zero u arcLen Function Returns the arc length Syntax arcLen Exp List variable start value end value Example To determine the ...

Страница 19: ...ator Exp List Example To extract the denominator of the fraction X 2 Y 1 2 CALC j EXTRCT b den a X c a Y bw Y 1 u numerator num Function Extracts the numerator of a fraction Syntax numerator Exp List Example To extract the numerator of the fraction X 2 Y 1 2 CALC j EXTRCT c num a X c a Y bw X 2 u gcd Function Returns the greatest common divisor Syntax gcd Exp List Exp List Example To determine the...

Страница 20: ...uation memory 3 3 EQUA c rclEqn c dw 3X Y 7 3X 6Y 63 The memory numbers of equations produced as the result of a recall are not updated u rclAllEqn rclAll Function Recall all eqn memory contents Syntax rclAllEqn The memory numbers of equations produced as the result of a recall are not updated u rewrite rewrit Function Moves the right side expression to the left side Syntax rewrite Eq Ineq List Ex...

Страница 21: ...X 3 to X and then substitute into 2X 3Y 5 3 EQUA g elim ca X da Y f a X a Y ca X dw 4Y 3 5 u getRight getRgt Function Gets the right side element Syntax getRight Eq Ineq List Example To extract the right side element ofY 2X2 3X 5 3 EQUA h getRgt a Y ca X x da X fw 2X2 3X 5 u invert Function Inverts two variables Syntax invert Exp Eq Ineq List variable name 1 variable name 2 If you omit the variabl...

Страница 22: ...cts two inequalities into a single expression Syntax andConnect Ineq 1 Ineq 2 Example To combine X 1 and X 3 into a single inequality 3 EQUA v andCon v3 EQUA b INEQUA b b v3 EQUA b INEQUA c dw 1 X 3 u eqn Function Recalls eqn memory contents Syntax eqn memory number Example To add 15 to both sides of the equation 6X 15 X 7 which is stored in equation memory 3 4 eqn d bfw 6X X 8 2X 3 9 or 2X 3 9 2 ...

Страница 23: ... w Example To clear the contents of variables X Y and Z 6 g 1 CLR b clrVar a X a Y aa Z w u clearVarAll VarAll Function Clears the contents of all 28 variables A to Z r θ Syntax clearVarAll 7 1 22 Using the CAS Computer Algebra System Mode 1When you start out with memories A B C and D for example and delete memories A and B the display shows only C D because they are the only memories remaining ...

Страница 24: ...mple To determine the minimum value of the elements in list 1 2 3 K1 LIST b CALC c Min b c d w 1 Example To compare each element of list 1 2 3 with the value 2 and produce a list whose elements are the minimum value resulting from each comparison K1 LIST b CALC c Min b c d cw 1 2 2 Example To compare the elements of list 1 2 3 and list 3 1 2 and produce a list whose elements are the minimum value ...

Страница 25: ...elements are the maximum value resulting from each comparison K1 LIST b CALC d Max b c d cw 2 2 3 Example To compare the elements of list 1 2 3 and list 3 1 2 and produce a list whose elements are the maximum value resulting from each comparison K1 LIST b CALC d Max b c d d b c w 3 2 3 u Mean Function Returns the mean of the elements in a list Syntax Mean List Mean List List The list must contain ...

Страница 26: ...le To determine the median of the elements in list 1 2 3 K1 LIST b CALC f Median b c d w 2 Example To determine the median of the elements in list 1 2 3 when their frequencies are 3 2 1 K1 LIST b CALC f Median b c d d c b w u Sum Function Returns the sum of the elements in a list Syntax Sum List The list must contain values or mathematical expressions only Equations and inequalities are not allowe...

Страница 27: ...t contain values or mathematical expressions only Equations and inequalities are not allowed Example To determine the cumulative frequency of the elements in list 1 2 3 K1 LIST b CALC i Cuml b c d w 1 3 6 u Percent Function Returns the percentage of each element in a list the sum of which is assumed to be 100 Syntax Percent List The list must contain values or mathematical expressions only Equatio...

Страница 28: ...iation of the elements in a list Syntax StdDev List The list must contain values or mathematical expressions only Equations and inequalities are not allowed Example To determine the sample standard deviation of the elements in list 1 2 4 K1 LIST b CALC l StdDev b c e w u Variance Vari Function Returns the variance of the elements in a list Syntax Variance List The list must contain values or mathe...

Страница 29: ...2 to List 1 Syntax Augment List List Example To combine list 1 2 and list 3 4 K1 LIST c CREATE c Augmnt b c d e w 1 2 3 4 u Fill Function Replaces the elements of a list with a specified value or expression This command can also be used to create a new list whose elements all contain the same value or expression Syntax Fill Exp Eq Ineq List Fill Exp numeric value Example To replace the elements of...

Страница 30: ...st must contain values or mathematical expressions only Equations and inequalities are not allowed Example To sort the elements of list 1 5 3 into descending order K1 LIST c CREATE f SortD b f d w 5 3 1 u SubList SubLst Function Extracts a specific section of a list into a new list Syntax SubList List start number end number Example To extract element 2 through element 3 from list 1 2 3 4 K1 LIST ...

Страница 31: ...le To convert list 3 5 and list 2 4 into a matrix K1 LIST d LIST b L Mat d f 3 2 c e w 5 4 u List Vect L Vect Function Converts a list into a vector Syntax List Vect List Example To convert list 3 2 into a vector K1 LIST d LIST c L Vect d c w 3 2 7 1 30 Using the CAS Computer Algebra System Mode ...

Страница 32: ... Dim b c d e f g w 2 3 u Det Function Returns the determinant of a matrix Syntax Det Mat Example To determine the determinant of the matrix below 1 2 4 5 K2 MAT b CALC c Det b c e f w 3 u Norm Function Returns the norm of a matrix Syntax Norm Mat Example To determine the norm of the matrix below 1 2 4 5 K2 MAT b CALC d Norm b c e f w 46 7 1 31 Using the CAS Computer Algebra System Mode 20011101 ...

Страница 33: ...le 0 894427191 0 4472135955 are the eigenvectors that correspond to 5 while 0 894427191 0 4472135955 are the eigenvectors that correspond to 1 An eigenvector has an infinite number of solutions The eigenvector displayed by this command is the one with a size of 1 u EigVl Function Returns the eigenvalue of a matrix Syntax EigVl Mat Example To determine the eigenvalue of the matrix below 3 4 1 3 K2 ...

Страница 34: ...9 5 2 4 4 K2 MAT b CALC g Rref c c a g b b j j f c e e w u Ref Function Returns the row echelon form of a matrix Syntax Ref Mat Example To determine the row echelon form of the matrix below 2 2 0 6 1 1 9 9 5 2 4 4 K2 MAT b CALC h Ref c c a g b b j j f c e e w 7 1 33 Using the CAS Computer Algebra System Mode 71 66 71 147 71 62 1 0 0 0 1 0 0 0 1 71 62 1 1 0 0 1 3 6 0 0 1 2 9 ...

Страница 35: ... 18 d i bi 0 4 16 av A al B w 0 0 1 The upper matrix is displayed as the calculation result To display the lower matrix recall the lower matrix variable A in this example specified by the command av A w To display the upper matrix recall the upper matrix variable B in this example specified by the command u Trn Function Transposes a matrix Syntax Trn Mat Example To transpose the matrix below 1 2 3...

Страница 36: ...e To create a 2 2 identity matrix K2 MAT c CREATE d Ident cw 1 0 0 1 u Fill Function Replaces the elements of a matrix with a specified value or expression This command can also be used to create a new matrix whose elements all contain the same value or expression Syntax Fill Exp Mat Fill Exp number of lines number of rows Example To replace the elements of the matrix below with X 3 4 1 2 K2 MAT c...

Страница 37: ...Mat Mat start row start column end row end column Example To extract the section from row 2 column 2 to row 3 column 3 from the following matrix 1 2 3 4 5 6 7 8 9 K2 MAT c CREATE f SubMat b c d e f g h i j 5 6 c c d dw 8 9 If you do not specify an end row and column all the entries from the start row column to the end of the matrix are extracted 7 1 36 Using the CAS Computer Algebra System Mode ...

Страница 38: ...umn of a matrix into a list Syntax Mat List Mat column number Example To convert column 2 of the matrix below to a list 1 2 3 4 K2 MAT d MAT b M List b c d e cw 2 4 u Mat Vect M Vect Function Converts a specific column of a matrix into a vector Syntax Mat Vect Mat column number Example To convert column 2 of the matrix below to a vector 1 2 3 4 K2 MAT d MAT c M Vect b c d e cw 2 4 7 1 37 Using the...

Страница 39: ... Syntax Row Exp Mat row number Example To multiply row 1 of the matrix below by X 1 2 3 4 K2 MAT e ROW c Row v b c X 2X d e bw 3 4 u Row Function Calculates the scalar product of one row of a matrix and adds the result to another row Syntax Row Exp Mat line number 1 line number 2 Example To multiply row 1 of the matrix below by X and add the result to row 2 1 2 3 4 K2 MAT e ROW d Row v b c 1 2 d e...

Страница 40: ...s one row of a matrix and to another row Syntax Row Mat row number 1 row number 2 Example To add row 1 of the matrix below to row 2 1 2 3 4 K2 MAT e ROW e Row b c 1 2 d e b cw 4 6 7 1 39 Using the CAS Computer Algebra System Mode ...

Страница 41: ... determine the cross product of vector 1 2 3 and vector 4 5 6 K3 VECT b CALC c CrossP b c d e f g w 3 6 3 u DotP Function Returns the dot product of two vectors Syntax DotP Vect Vect Example To determine the dot product of vector 1 2 3 and vector 4 5 6 K3 VECT b CALC d DotP b c d e f g w 32 u Norm Function Returns the norm of a vector Syntax Norm Vect Example To determine the norm of the vector 1 ...

Страница 42: ...ad K3 VECT b CALC g Angle b c d e w u Augment Augmnt Function Combines two vectors Syntax Angle Vect Vect Example To combine vector 1 2 and vector 3 4 K3 VECT c CREATE b Augmnt b c d e w 1 2 3 4 u Fill Function Replaces the elements of a vector with a specified value or expression Syntax Fill Exp Vect Example To replace the components of the vector below with X K3 VECT c CREATE c Fill v d e w X X ...

Страница 43: ...le To convert vector 3 2 into a list K3 VECT d VECT b V List d c w 3 2 u Vect Mat V Mat Function Converts vectors into a matrix Syntax Vect Mat Vect Vect Example To convert vector 3 5 and 2 4 into a matrix K3 VECT d VECT c V Mat d f 3 2 c e w 5 4 7 1 42 Using the CAS Computer Algebra System Mode ...

Страница 44: ...ions Also the following commands are available in the Algebra Mode only u arrange arrang Function Collects like terms and arranges them in order starting with the term that contains the smallest coefficient Syntax arrange Exp Eq Ineq Example To arrange 2X 3 5X 8Y in sequence of its variables 1 TRNS j arrang ca X d fa X ia Y w 5X 2X 8Y 3 u replace replac Function Replaces a variable with the expres...

Страница 45: ...torial Mode displays a menu of the following expression types Linear Equation Linear Inequality Quadratic Equation Simul Simultaneous Equation Use the cursor keys to highlight the expression type you want to specify and then press w This displays a list of formulas for the expression type you select Move the cursor to the formula you want to use In the case of Linear Inequality press 4 TYPE to sel...

Страница 46: ... F AX B C Quadratic Equation 5 Types AX2 B AX B 2 C AX2 BX C 0 AX2 BX C D AX2 BX C DX2 EX F Simul Equation 10 Types AX BY C Y AX B DX EY F Y CX D AX BY C 0 AX BY C DX EY F DX EY F 0 GX HY I JX KY L AX BY C AX BY C Y DX E DX EY F 0 AX BY C AX BY C 0 DX EY F GX HY I Y DX E AX BY C DX EY F AX BY C 0 Y GX H DX EY F GX HY I Pressing 6 EXCH reverses the left side and right side elements of the expressio...

Страница 47: ...ress 6 EXE to define the coefficient 3 SMPL displays a number of preset sample expressions Highlight the one you want to use and then press w to define it Pressing4 SEED displays a number selection screen When you want to create the same problem on another calculator specify an appropriate matching number and press w No matter what method you use the expression you define is displayed in the outpu...

Страница 48: ...has two solutions the left side and right side are obtained for the value where the pointer is located In the case of simultaneous equations where the left side and right side of the second equation are dissimilar even though the left side and right side of the first equation match the left side and right side of the second equation only are obtained In other cases the left side and right side of ...

Страница 49: ...401 7 3 5 Tutorial Mode You can press 4 MANU to change to the Manual Mode or 5 AUTO to change to the Auto Mode Example To solve 4X 8 in the Verify Mode Linear Equation AX B 2 INPUT ewiw6 EXE 4 VRFY cw 6 JUDG ...

Страница 50: ...n is the same as that in the Algebra Mode After you obtain a result you can press 5 JUDG to determine whether or not it is correct DISP Determines whether the expression in the display area is a correct solution Identi identity of left side and right side Many many solutions No sol no solution You can press 6 AUTO to change to the Auto Mode Example Solve 4X 8 in the Manual Mode Linear Equation AX ...

Страница 51: ...xample 4X2 16 True X 2 X 2 Besides TRUE the messages shown below can also appear as the result of verification CAN NOT JUDGE appears in the Manual Mode while the other messages appear in both the Verify Mode and Manual Mode 20010102 ...

Страница 52: ...sides of the expression by the same value to isolate the coefficient X or Y As with the Algebra Mode the screen is divided between an input area and a display area Each press of 6 NEXT advances to the next step 6 NEXT is not shown on the display when the solution is obtained You can scroll back through the steps by pressing 1 BACK Example To solve 4X 8 in the Auto Mode Linear Equation AX B 2 INPUT...

Страница 53: ...lure of a result to appear immediately does not indicate malfunction of the computer Any expression can be displayed in various different formats Because of this you should not assume that an expression is wrong just because it does not appear as you expected This calculator performs integration calculations under the assumption that integrals are always positive even when the integrals switch bet...

Страница 54: ...20010102 ...

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