Simplex Tableaux:
Simplex Tableaux for linear programing problems can also be entered, since
MAT offers step by step tutorials on the Simplex method. The row
corresponding to the objective function should be separated from the rows representing
the constraints for the problem, by the use of a horizontal line separator. The objective
row should always be entered as the last row.
For example, to enter the Simplex Tableau for the following problem:
The user types: [2,5,1,0,0|58;-1,1,0,1,0,-15_-20,-70,0,0,1,0]. Where the
"underscore" symbol "_" instead of the semicolon, indicates
that the next row is the objective row.
The tableau displayed by MAT will look as follows:
Matrix Archive:
The matrix editing window allows the user to archive the matrix that was just
typed, by simply clicking on the button : Add to Archive. All the matrices
stored in said archive, can be accessed and brought to the screen for future use.
While in the matrix editing window, any matrix stored in the archive can be retrieved
by pressing the View Archive button and clicking on the selected matrix expression
listed in drop-down menu of the archive, and using the option Use Selected.
Editing Matrices:
The simplest and quickest way of changing the values of elements in a matrix is by
right mouse button clicking on the element to be changed and by using the up/down
arrows on the spin control window that is prompted, to increase/decrease the numerical
values.
If the changes involve the typing of algebraic expressions or if the actual dimensions
of the matrix have to be modified, the editing box for the particular matrix should be accessed.
This is done through:
(a) The general MAT menu (accessed by left mouse button clicking
on the background) - option: Edit Expression (an element in the relevant
matrix should be previously highlighted to indicate which is the selected matrix, otherwise
the very last matrix entered would be edited).
(b) The matrix handling menu for the matrix (accessed by right mouse
button clicking on the region of the screen limited by the matrix's OPENING bracket) -
option: Edit Matrix
Once the editing window is accessed, the relevant changes should be made on the
edit box (upper part of the window), and the changes would be instantly refreshed on the
lower part of the window. When the changes are done, the "Enter" key should be pressed or
the Done button clicked.
The Hidden Menus :
Once at least one matrix is displayed on the MAT screen, two other menus can
be accessed by the user. One gives the user options on general handling of a particular
matrix, and the other one shows options with reference to different operations to be
performed within the matrix, these include diagonalization,
calculation of the determinant for a square matrix, transposition, simplex method for
a simplex tableau, raw and column multiplication by scalars, addition or subtraction of
columns or rows, etc.
Depending on where on the screen the user clicks, and with which mouse button, the
diferent menus will pop up. When a matrix is displayed on the screen, new
"clicking spaces" are defined. This is illustrated in the following picture:
Matrix Handling Menu
It is accessed by RIGHT mouse button clicking on the opening bracket area of the matrix
of interest. The menu options are:
Edit Expression | To access
the editing window for the matrix. |
Delete Expression | To
delete from the screen the matrix expression. |
Clone Expression |
Creates an exact duplicate of the matrix on the screen. |
Add Expression to Archive |
Copies the matrix expression to the matrix archive for future use. |
Expression Audit Trail |
When this option is checked (by clicking on it) a copy
of the original matrix expression is kept on the screen even after matrix operations
have been performed. |
Matrix Operations Menu
It is accessed by RIGHT mouse button clicking on the closing bracket area of the matrix
of interest. This menu offers operations within a unique matrix expression.
The menu options are:
Diagonalize Matrix |
Performs the diagonalization of the matrix (ideal for Gauss-Jordan
method using augmented matrices). This can be done in ONE step by clicking on the
sub-option: Just Do It, or through a step-by-step tutorial, by clicking on
the Animate sub-option. |
Transpose |
Creates the transpose of the given matrix |
Evaluate Simplex |
Only shown (available) when the matrix in question is in
Simplex
Tableau form. Performs the Simplex maximization method in ONE step (by clicking on
Just Do It, or through the step-by-step tutorial (by clicking on the Animate
sub-option) |
Compute Determinant |
Computes the determinant of a SQUARE matrix. Nothing will be performed if the matrix in
question is not a square matrix. |
Scale Row |
Multiplies the user's selected row by a scalar, throught an interactive
dialog process. |
Scale Column |
Multiplies the user's selected column by a scalar, throught an interactive
dialog process. |
Exchange Rows |
Exchanges the rows selected by the user using interactive dialog boxes. |
Exchange Columns |
Exchanges the columns selected by the user using interactive dialog boxes. |
Subtract Rows |
Subtracts a row multiplied by a given scalar, from another row, using dialog boxes. |
Subtract Columns |
Subtract a column multiplied by a given scalar, from another column, using dialog boxes. |
Operations Between Matrices:
Addition, subtraction and multiplication of matrices should be typed as such when
adding a new expression. Some examples follow:
To add the matrices:
after accessing the edit window through the Add Expression option in the main menu,
the addition is typed in the edit box as shown in the following figure:
The operation will appear on the MAT screen as follows:
To multiply the matrices:
after accessing the edit window through the Add Expression option in the main menu,
the multiplication is typed in the edit box as shown in the following figure:
The operation will appear on the MAT screen as follows:
Tutorials :
MAT offers tutorials for lengthy or complicated methods, to facilitate the
teaching and learning of the subject. Tutorials are available for Gauss-Jordan
diagonalization and for the Simplex method. Once the appropriate matrix, augmented matrix or
Simplex tableau has been typed by the user, when accessing either Diagonalize Matrix
or Evaluate Simplex, two sub-options are available. These are: Just Do It, or
Animate. The step-by-step tutorials are accessed by clicking on the Animate
option. Once in the "Animator" window, the user advances step by step clicking on
the One Step button. The Animator window shows what the next step would be and
the result of each step.
The following figure illustrates one step in a Simplex evaluation:
Algebraic Expressions :
MAT allows the use of generic algebraic notation to demonstrate
basic properties of operations between matrices. Characters can be entered as matrix
elements instead of numerical values, and the resulting operations to be performed
will be shown on the MAT screen.
For example, to illustrate the multiplication of a generic (2 x 3) matrix by a
generic (3 x 2) matrix, the following can be entered through the Add Expression
option of the main menu:
The result of the multiplication is shown on the MAT screen as
follows: