Casio FX-9750G 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 product 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

92
2 (row) × 2 (column) matrix
123456
Not dimension preset
6-1 Before Performing Matrix Calculations
In the Main Menu, select the MAT icon and press wto enter the Matrix Mode and
display its initial screen.
1(DEL) ....... Delete specific matrix
2(DEL•A).... Delete all matrices
• The maximum matrix dimension (size) is 255 (rows) ×255 (columns).
kk
kk
kAbout Matrix Answer Memory (MatAns)
The calculator automatically store matrix calculation results in MatrixAnswer 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 MatrixAnswer 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
Specify the number of rows.
cw
Specify the number of columns.
d
w
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93
• 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 dimen-
sions, 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 inte-
gers up to five digits (one digit used for 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).
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).
1(DEL)
Highlighted cell (up to six digits
can be displayed)
Value in currently highlighted cell
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Before Performing Matrix Calculations 6 - 1

94
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(DEL•A)
2. Press 1(YES) to delete all matrices in memory or 6(NO) to abort the opera-
tion without deleting anything.
• The indicator “None” is shown for all the matrices.
123456
6 - 1 Before Performing Matrix Calculations

95
6-2 Matrix Cell Operations
You can perform any of the following operations involving the cells of a matrix on the
display.
• Row swapping, scalar product, addition
• Row deletion, insertion, addition
• Column deletion, insertion, addition
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 w.12
MatrixA = 34
56
1(R•OP) ..... Row calculation menu
2(ROW)...... Row operation menu
3(COL) ....... 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.
1(R•OP)
1(Swap) ..... Row swap
2(xRw) ....... Scalar product for a specific row
3(xRw+) ..... Addition of scalar product of specific row to another row
4(Rw+) ....... Addition of contents of specific row to another row
uu
uu
uTo swap two rows
Example To swap rows two and three of the following matrix :
12
Matrix A = 34
56
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96
1(R•OP)1(Swap)
Input the number of the rows you want to swap.
cw
dw
uu
uu
uTo calculate the scalar product of a row
Example To calculate the scalar product of row 2 of the following matrix by 4 :
12
Matrix A = 34
56
1(R•OP)
2(×Rw)
Input multiplier value.
ew
Specify row number.
cw
uu
uu
uTo calculate the scalar product of a row and add the result to
another row
Example To calculate the scalar product of row 2 of the following matrix
by 4 and add the result to row 3 :
12
Matrix A = 34
56
1(R•OP)
3(×Rw+)
Input multiplier value.
ew
Specify number of row whose scalar product
should be calculated.
cw
Specify number of row where result should be
added.
dw
6 - 2 Matrix Cell Operations

97
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 number of row to be added.
cw
Specify number of row to be added to.
dw
kk
kk
kRow Operations
The following menu appears whenever you press 2(ROW) while a recalled matrix
is on the display.
2(ROW)
1(DEL) ....... Delete row
2(INS) ........ Insert row
3(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)
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Matrix Cell Operations 6 - 2

98
uu
uu
uTo insert a row
Example To insert a new row between rows one and two of the following
matrix :
12
Matrix A = 34
56
2(ROW)c
2(INS)
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)
123456
123456
6 - 2 Matrix Cell Operations

99
kk
kk
kColumn Operations
The following menu appears whenever you press 3(COL) while a recalled matrix
is on the display.
3(COL)
1(DEL) ....... Delete column
2(INS) ........ Insert column
3(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)
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
123456
123456
123456
Matrix Cell Operations 6 - 2

100
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)
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6 - 2 Matrix Cell Operations

101
6-3 Modifying Matrices Using Matrix Commands
In addition to using the MATRIX list to create and modify a matrix, you can also use
matrix commands to input data and create a matrix without actually displaying it.
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.
K2(MAT)
The following describes only the matrix command menu items that are used for
creating matrices and inputting matrix data.
1(Mat) ........ Mat command (matrix specification)
2(M→L)...... Mat→List command (assign contents of selected column to
list file)
5(Aug) ........ Augment command (link two matrices)
6(g) ........... Next menu
6(g)
1(Iden) ....... Identity command (identity matrix input)
2(Dim) ........ Dim command (dimension check)
3(Fill) .......... Fill command (identical cell values)
6(g) ........... Previous menu
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 Athrough Z]
• The maximum value of both mand nis 255.
123456
123456
P.105
P.31

102
Example 1 To input the following data as Matrix A :
135
246
K2(MAT)
![![b,d,f
!]![c,e,g
!]!]a1(Mat)aA
w
• An error (Mem 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)da
Number of rows/columns
6(g)1(Mat)aA
w
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)
aA
123456
P.101
123456
123456
6 - 3 Modifying Matrices Using Matrix Commands
P.101
Matrix name

103
w
The display shows that MatrixA consists of two rows and three columns.
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, orAns)
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
Modifying Matrices Using Matrix Commands 6 - 3
123456
123456
Number of rows
Number of columns
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104
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 and theAugment command (5) to combine two ex-
isting 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,
Filler value
6(g)1(Mat)aA
w
Example 2 To combine the following two matrices :
A = 1B = 3
24
K2(MAT)
5(Aug)1(Mat)aA,
1(Mat)aB
w
• The two matrices you combine must have the same number of rows. An error
(Ma ERROR) occurs if you try to combine two matrices that have different num-
bers 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
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6 - 3 Modifying Matrices Using Matrix Commands
P.101
P.101

105
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 variable names Athrough Z, andnis any value.The
above does not affect the contents of Matrix Answer Memory.
Modifying Matrices Using Matrix Commands 6 - 3
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106
6-4 Matrix Calculations
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.
K2(MAT)
The following describes only the matrix commands that are used for matrix arithme-
tic operations.
1(Mat) ........ Mat command (matrix specification)
3(Det)......... Det command (determinant command)
4(Trn) ......... Trn command (transpose matrix command)
6(g) ........... Next menu
6(g)
1(Iden) ....... Identity command (identity matrix input)
6(g) ........... Previous menu
All of the following examples assume that matrix data is already stored in memory.
kk
kk
kMatrix Arithmetic Operations
The following is the format for matrix arithmetic operations.
Matrix 1 Arithmetic operator key Matrix 2
MatA +MatA
-w
Mat Z *Mat Z
MatAns MatAns
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107
Example 1 To add the following two matrices (Matrix A + Matrix B) :
A = 11 B = 23
21 21
1(Mat)aA+
1(Mat)aB
w
This display indicates the following result.
A + B = 34
42
Example 2 To multiply the two matrices in Example 1 (MatrixA ×Matrix B)
1(Mat)aA*
1(Mat)aB
w
This display indicates the following result.
A ×B = 44
67
• The two matrices must have the same dimensions in order to be added or sub-
tracted. An error (Dim 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 (Dim 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 (1) command to
input the identity matrix.
123456
123456
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108
Example 3 To multiply Matrix A (from Example 1) by a 2 ×2 identity matrix
1(Mat)aA*
6(g)1(Iden)c
Number of rows and columns.
w
This display indicates the following result.
A ×E = 11
21
kk
kk
kMatrix Scalar Product
The following is the format for calculating a matrix scalar product, which multiplies
the value in each cell of the matrix by the same value.
Scalar value Matrix
MatA
kw
Mat Z
MatAns
Example Calculate the scalar product of the following matrix using a
multiplier value of 4 :
Matrix A = 12
34
e1(Mat)aA
w
This display indicates the following result.
4A = 48
12 16
123456
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6 - 4 Matrix Calculations

109
kk
kk
kDeterminant
The following is the format for obtaining a determinant.
Matrix
MatA
3(Det) w
Mat Z
MatAns
Example Obtain the determinant for the following matrix :
123
Matrix A = 456
–1 –2 0
3(Det)1(Mat)aAw
This display indicates determinant |A| = –9.
• 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 pro-
duces an error (Dim ERROR).
• The determinant of a 2 ×2 matrix is calculated as shown below.
| A | = a11 a12 = a11 a22 – 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
= a11 a22a33 + a12a23a31 + a13a21a32
– a11a23a32 – a12a21a33 – a13a22a31
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Matrix Calculations 6 - 4

110
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
MatA
4(Trn) w
Mat Z
MatAns
Example To transpose the following matrix:
12
Matrix A = 34
56
4(Trn)1(Mat)aA
w
This operation produces the following result.
At= 135
246
kk
kk
kMatrix Inversion
The following is the format for matrix inversion.
Matrix
MatA
!X w
Mat Z
MatAns
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6 - 4 Matrix Calculations