Casio CFX-9850G PLUS User manual

Matrix Calculations
26 matrix memories (Mat A through Mat Z) plus a Matrix Answer
Memory (MatAns), make it possible to perform the following
matrix operations.
•Addition, subtraction, multiplication
•Scalar multiplication calculations
•Determinant calculations
•Matrix transposition
•Matrix inversion
•Matrix squaring
•Raising a matrix to a specific power
•Absolute value, integer part extraction, fractional part extraction,
maximum integer calculations
•Matrix modification using matrix commands
6-1 Before Performing Matrix Calculations
6-2 Matrix Cell Operations
6-3 Modifying Matrices Using Matrix Commands
6-4 Matrix Calculations
6
Chapter

80
2 (row) × 2 (column) matrix
6-1 Before Performing Matrix Calculations
In the Main Menu, select the MAT icon to enter the Matrix Mode and display its
initial screen.
• {DEL}/{DEL·A} ... deletes {a specific matrix}/{all matrices}
•The maximum number of rows that can be specified for a matrix is 255, and the
maximum number of columns is 255.
kk
kk
kAbout Matrix Answer Memory (MatAns)
The calculator automatically stores matrix calculation results in Matrix Answer
Memory. Note the following points about Matrix Answer Memory.
•Whenever you perform a matrix calculation, the current Matrix Answer Memory
contents are replaced by the new result. The previous contents are deleted and
cannot be recovered.
•Inputting values into a matrix does not affect Matrix Answer Memory contents.
kk
kk
kCreating a Matrix
To create a matrix, you must first define its dimensions (size) in the MATRIX list.
Then you can input values into the matrix.
uu
uu
uTo specify the dimensions of a matrix
Example To create a 2-row ×3-column matrix in the area named Mat B
Highlight Mat B.
c
Not dimension preset
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81
Specify the number of rows.
cw
Specify the number of columns.
d
w
•All of the cells of a new matrix contain the value 0.
•If “Mem ERROR” remains next to the matrix area name after you input the
dimensions, it means there is not enough free memory to create the matrix you
want.
uu
uu
uTo input cell values
Example To input the following data into Matrix B :
123
456
Select Mat B.
c
w
bwcwdw
ewfwgw
(Data is input into the highlighted cell.
Each time you press w, the highlight-
ing move to the next cell to the right.)
•Displayed cell values show positive integers up to six digits, and negative
integers up to five digits (one digit used for the negative sign). Exponential
values are shown with up to two digits for the exponent. Fractional values are
not displayed.
•You can see the entire value assigned to a cell by using the cursor keys to
move the highlighting to the cell whose value you want to view.
•The amount of memory required for a matrix is ten bytes per cell. This means
that a 3 ×3 matrix requires 90 bytes of memory (3 ×3 ×10 = 90).
Before Performing Matrix Calculations 6 - 1
Highlighted cell (up to six digits
can be displayed)
Value in currently highlighted cell

82
kk
kk
kDeleting Matrices
You can delete either a specific matrix or all matrices in memory.
uu
uu
uTo delete a specific matrix
1. While the MATRIX list is on the display, use fand cto highlight the matrix
you want to delete.
2. Press 1(DEL).
3. Press 1(YES) to delete the matrix or 6(NO) to abort the operation without
deleting anything.
•The indicator “None” replaces the dimensions of the matrix you delete.
uu
uu
uTo delete all matrices
1. While the MATRIX list is on the display, press 2(DEL·A).
2. Press 1(YES) to delete all matrices in memory or 6(NO) to abort the
operation without deleting anything.
•The indicator “None” is shown for all the matrices.
6 - 1 Before Performing Matrix Calculations

83
6-2 Matrix Cell Operations
Use the following procedure to prepare a matrix for cell operations.
1. While the MATRIX list is on the display, use fand cto highlight the name
of the matrix you want to use.
2. Press wand the function menu with the following items appears.
• {R·OP} ... {row calculation menu}
• {ROW}/{COL} ... {row}/{column} operation menu
All of the following examples use Matrix A recalled by the above operation.
kk
kk
kRow Calculations
The following menu appears whenever you press 1(R·OP) while a recalled
matrix is on the display.
• {Swap} ... {row swap}
• {×Rw} ... {product of specified row and scalar}
• {×Rw+} ... {addition of one row and the product of a specified row with a scalar}
• {Rw+} ... {addition of specified row to another row}
uu
uu
uTo swap two rows
Example To swap rows 2 and 3 of the following matrix :
12
Matrix A = 34
56
1(R·OP)1(Swap)
Input the number of the rows you want to swap.
cwdw

84
uu
uu
uTo calculate the product of a row
Example To calculate the product of row 2 of the following matrix and the
scalar 4 :
12
Matrix A = 34
56
1(R·OP)2(×Rw)
Input multiplier value.
ew
Specify row number.
cw
uu
uu
uTo calculate the product of a row and add the result to another row
Example To calculate the product of row 2 of the following matrix and
the scalar 4, then add the result to row 3 :
12
Matrix A = 34
56
1(R·OP)3(×Rw+)
Input multiplier value.
ew
Specify the row number whose product should
be calculated.
cw
Specify the row number where result should be
added.
dw
uu
uu
uTo add two rows together
Example To add row 2 to row 3 of the following matrix :
12
Matrix A = 34
56
1(R·OP)4(Rw+)
Specify the row number to be added.
cw
Specify the row number to be added to.
dw
6 - 2 Matrix Cell Operations

85
kk
kk
kRow Operations
The following menu appears whenever you press 2(ROW) while a recalled
matrix is on the display.
• {DEL} ... {delete row}
• {INS} ... {insert row}
• {ADD} ... {add row}
uu
uu
uTo delete a row
Example To delete row 2 of the following matrix :
12
Matrix A = 34
56
2(ROW)c
1(DEL)
uu
uu
uTo insert a row
Example To insert a new row between rows 1 and 2 of the following
matrix :
12
Matrix A = 34
56
2(ROW)c
2(INS)
Matrix Cell Operations 6 - 2

86
uu
uu
uTo add a row
Example To add a new row below row 3 of the following matrix :
12
Matrix A = 34
56
2(ROW)cc
3(ADD)
kk
kk
kColumn Operations
The following menu appears whenever you press 3(COL) while a recalled
matrix is on the display.
• {DEL} ... {delete column}
• {INS} ... {insert column}
• {ADD} ... {add column}
uu
uu
uTo delete a column
Example To delete column 2 of the following matrix :
12
Matrix A = 34
56
3(COL)e
1(DEL)
6 - 2 Matrix Cell Operations

87
uu
uu
uTo insert a column
Example To insert a new column between columns 1 and 2 of the
following matrix :
12
Matrix A = 34
56
3(COL)e
2(INS)
uu
uu
uTo add a column
Example To add a new column to the right of column 2 of the following
matrix :
12
Matrix A = 34
56
3(COL)e
3(ADD)
Matrix Cell Operations 6 - 2

88
6-3 Modifying Matrices Using Matrix Commands
[OPTN]-[MAT]
uu
uu
uTo display the matrix commands
1. From the Main Menu, select the RUN icon and press w.
2. Press Kto display the option menu.
3. Press 2(MAT) to display the matrix operation menu.
The following describes only the matrix command menu items that are used for
creating matrices and inputting matrix data.
• {Mat} ... {Mat command (matrix specification)}
• {M→L} ... {Mat→List command (assign contents of selected column to list file)}
• {Aug} ... {Augment command (link two matrices)}
• {Iden} ... {Identity command (identity matrix input)}
• {Dim} ... {Dim command (dimension check)}
• {Fill} ... {Fill command (identical cell values)}
kk
kk
kMatrix Data Input Format
The following shows the format you should use when inputting data to create a
matrix using the matrix operation menu’s Mat command.
a11 a12 a1n
a21 a22 a2n
am1am2amn
= [ [a11, a12, ..., a1n] [a21, a22, ..., a2n] .... [am1, am2, ..., amn] ]
→Mat [letter A through Z]
•The maximum value of both mand nis 255.
Example 1 To input the following data as Matrix A :
135
246
K2(MAT)
![![b,d,f
!]![c,e,g
!]!]a1(Mat)aA
P. 2 7
P. 9 1

89
w
•An error occurs if memory becomes full as you are inputting data.
•You can also use the above format inside a program that inputs matrix data.
uu
uu
uTo input an identity matrix
Use the matrix operation menu’s Identity command (1) to create an identity
matrix.
Example 2 To create a 3 ×3 identity matrix as Matrix A
K2(MAT)6(g)1(Iden)
da6(g)1(Mat)aAw
Number of rows/columns
uu
uu
uTo check the dimensions of a matrix
Use the matrix operation menu’s Dim command (2) to check the dimensions of
an existing matrix.
Example 3 To check the dimensions of Matrix A, which was input in
Example 1
K2(MAT)6(g)2(Dim)6(g)
1(Mat) aAw
The display shows that Matrix A consists of two rows and three columns.
You can also use {Dim} to specify the dimensions of the matrix.
Example 4 To specify dimensions of 2 rows and 3 columns for Matrix B
!{c,d!}aK
2(MAT)6(g)2(Dim)6(g)
1(Mat)aBw
Modifying Matrices Using Matrix Commands 6 - 3
Matrix name
Number of rows
Number of columns

90
kk
kk
kModifying Matrices Using Matrix Commands
You can also use matrix commands to assign values to and recall values from an
existing matrix, to fill in all cells of an existing matrix with the same value, to
combine two matrices into a single matrix, and to assign the contents of a matrix
column to a list file.
uu
uu
uTo assign values to and recall values from an existing matrix
Use the following format with the matrix operation menu’s Mat command (1) to
specify a cell for value assignment and recall.
Mat X [m, n]
X..................... matrix name (A through Z, or Ans)
m..................... row number
n...................... column number
Example 1 Assign 10 to the cell at row 1, column 2 of the following matrix :
12
Matrix A = 34
56
baaK2(MAT)1(Mat)
aA![b,c!]w
Example 2 Multiply the value in the cell at row 2, column 2 of the above
matrix by 5
K2(MAT)1(Mat)
aA![c,c!]
*fw
uu
uu
uTo fill a matrix with identical values and to combine two matrices
into a single matrix
Use the matrix operation menu’s Fill command (3) to fill all the cells of an
existing matrix with an identical value, or the Augment command (5) to combine
two existing matrices into a single matrix.
Example 1 To fill all of the cells of Matrix A with the value 3
K2(MAT)6(g)3(Fill)
d,6(g)1(Mat)aAw
Filler value
6 - 3 Modifying Matrices Using Matrix Commands

91
Example 2 To combine the following two matrices :
A = 1B = 3
24
K2(MAT)5(Aug)1(Mat)
aA,1(Mat)aBw
•The two matrices you combine must have the same number of rows. An error
occurs if you try to combine two matrices that have different numbers of rows.
uu
uu
uTo assign the contents of a matrix column to a list file
Use the following format with the matrix operation menu’s Mat→List command
(2) to specify a column and a list file.
Mat → List (Mat X, m) →List n
X = matrix name (A through Z, or Ans)
m= column number
n= list number
Example To assign the contents of column 2 of the following matrix to list
file 1 :
12
Matrix A = 34
56
K2(MAT)2(M→L)1(Mat)
aA,c)a
Column number
K1(LIST)1(List)bw
You can use Matrix Answer Memory to assign the results of the above matrix
input and edit operations to a matrix variable. To do so, use the following
syntax.
•Fill (n, Mat α ) →Mat β
•Augment (Mat α , Mat β ) →Mat γ
In the above, α , β , and γ are any variable names A through Z, and nis any
value.
The above does not affect the contents of Matrix Answer Memory.
Modifying Matrices Using Matrix Commands 6 - 3

92
6-4 Matrix Calculations [OPTN]-[MAT]
Use the matrix command menu to perform matrix calculation operations.
uu
uu
uTo display the matrix commands
1. From the Main Menu, select the RUN icon and press w.
2. Press Kto display the option menu.
3. Press 2(MAT) to display the matrix command menu.
The following describes only the matrix commands that are used for matrix
arithmetic operations.
• {Mat} ... {Mat command (matrix specification)}
• {Det} ... {Det command (determinant command)}
• {Trn} ... {Trn command (transpose matrix command)}
• {Iden} ... {Identity command (identity matrix input)}
All of the following examples assume that matrix data is already stored in memory.
kk
kk
kMatrix Arithmetic Operations
Matrix 1 Arithmetic operator key Matrix 2
Mat A +Mat A
-w
Mat Z *Mat Z
MatAns MatAns
Example 1 To add the following two matrices (Matrix A + Matrix B) :
A = 11 B = 23
21 21
1(Mat)aA+
1(Mat)aBw
Example 2 To multiply the two matrices in Example 1 (Matrix A ×Matrix B)
1(Mat)aA*
1(Mat)aBw
P. 2 7

93
•The two matrices must have the same dimensions in order to be added or
subtracted. An error occurs if you try to add or subtract matrices of different
dimensions.
•For multiplication, the number of columns in Matrix 1 must match the number of
rows in Matrix 2. Otherwise, an error occurs.
•You can use an identity matrix in place of Matrix 1 or Matrix 2 in the matrix
arithmetic format. Use the matrix command menu’s Identity command (1)
to input the identity matrix.
Example 3 To multiply Matrix A (from Example 1) by a 2 ×2 identity matrix
1(Mat)aA*
6(g)1(Iden)cw
Number of rows and columns.
kk
kk
kMatrix Scalar Multiplication
The following is the format for calculating a matrix scalar multiplication, which
multiplies the value in each cell of the matrix by the same value.
Scalar value Matrix
Mat A
kw
Mat Z
MatAns
Example Calculate the product of the following matrix using a multiplier
value of 4 :
Matrix A = 12
34
e1(Mat)aAw
kk
kk
kDeterminant
Matrix
Mat A
3(Det) w
Mat Z
MatAns
Matrix Calculations 6 - 4

94
Example Obtain the determinant for the following matrix :
123
Matrix A = 456
–1 –2 0
3(Det)1(Mat)aAw
•Determinants can be obtained only for square matrices (same number of rows
and columns). Trying to obtain a determinant for a matrix that is not square
produces an error.
•The determinant of a 2 ×2 matrix is calculated as shown below.
|A|= a11 a12 = a11a22 – a12a21
a21 a22
•The determinant of a 3 ×3 matrix is calculated as shown below.
a11 a12 a13
|A|= a21 a22 a23
a31 a32 a33
= a11a22a33 + a12a23a31 + a13a21a32
– a11a23a32 – a12a21a33 – a13a22a31
kk
kk
kMatrix Transposition
A matrix is transposed when its rows become columns and its columns become
rows. The following is the format for matrix transposition.
Matrix
Mat A
4(Trn) w
Mat Z
MatAns
Example To transpose the following matrix :
12
Matrix A = 34
56
4(Trn)1(Mat)aAw
6 - 4 Matrix Calculations

95
kk
kk
kMatrix Inversion
Matrix
Mat A
!X w
Mat Z
MatAns
Example To invert the following matrix :
Matrix A = 12
34
1(Mat)aA!Xw
•Only square matrices (same number of rows and columns) can be inverted.
Trying to invert a matrix that is not square produces an error.
•Amatrix with a value of zero cannot be inverted. Trying to invert a matrix with
value of zero produces an error.
•Calculation precision is affected for matrices whose value is near zero.
•Amatrix being inverted must satisfy the conditions shown below.
A A–1 = A–1 A = E = 10
01
•The following shows the formula used to invert Matrix A into inverse matrix
A–1.
A = ab
cd
A–1 = 1d–b
ad – bc –c a Note that ad – bc G0.
Matrix Calculations 6 - 4

96
kk
kk
kSquaring a Matrix
Matrix
Mat A
xw
Mat Z
MatAns
Example To square the following matrix :
Matrix A = 12
34
1(Mat)aAxw
kk
kk
kRaising a Matrix to a Power
Matrix Natural number
Mat A
Mkw
Mat Z
MatAns
Example To raise the following matrix to the third power :
Matrix A = 12
34
1(Mat)aAMdw
kk
kk
kDetermining the Absolute Value, Integer Part, Fraction
Part, and Maximum Integer of a Matrix
Function command Matrix
Abs Mat A
Frac w
Int Mat Z
Intg MatAns
6 - 4 Matrix Calculations

97
Example To determine the absolute value of the following matrix :
Matrix A = 1–2
–3 4
K6(g)4(NUM)1(Abs)
K2(MAT)1(Mat)aAw
•Determinants and inverse matrices are calculated using the elimination
method, so errors (such as dropped digits) may be generated.
•Matrix operations are performed individually on each cell, so calculations
may require considerable time to complete.
•The calculation precision of displayed results for matrix calculations is ±1 at
the least significant digit.
•If a matrix calculation result is too large to fit into Matrix Answer Memory, an
error occurs.
•You can use the following operation to transfer Matrix Answer Memory
contents to another matrix (or when Matrix Answer Memory contains a
determinant to a variable).
MatAns →Mat α
In the above, α is any variable name A through Z. The above does not affect
the contents of Matrix Answer Memory.
Matrix Calculations 6 - 4

98