Working with Matrices

You can use TEMATH's Matrix Calculator to do matrix arithmetic, solve systems of linear equations, perform elementary row or column operations, find the inverse of a matrix, find the eigenvalues and eigenvectors of a matrix, calculate the norm, determinant, and trace of a matrix, and much more.

Basic Matrix Calculator Operations

1. Entering a Matrix

Select Calculators - Matrix Calculator... from the Work menu.
Double-click the Resize command, enter the number of rows and the number of columns of the matrix, and
Click the enter button or press the Enter key.
Enter the values of the matrix. You can move around the matrix by using the Enter, Return, Tab, and Arrow keys.

2. Selecting a Row or Column of a Matrix

To select a row or column,

Move the mouse until the cursor is between the label and the matrix.
The cursor will change its shape to a small right-pointing arrow for rows and to a small downward-pointing arrow for columns.
Click the mouse button to highlight (select) the row or column.

3. Deleting Rows or Columns from a Matrix

Select the row or column that you want deleted, then
Select Delete Row (Column) from the Edit menu.

4. Storing and Recalling Matrices

To temporarily store a matrix, click the store button and then click one of the memory buttons A, B, C orD.
To recall a stored matrix, click the recall button and then one of the memory buttons.

5. Canceling a Matrix Arithmetic Operation

If you make a mistake and wish to cancel an operation, click the ce button.

6. Saving (Adding) a Matrix to the Work window

Enter your matrix into the Matrix Calculator window.
Select Add Matrix from the Work menu.

Matrix Arithmetic

We'll start by entering three matrices A, B, and C and storing them into the memory location with the same name.

To enter these matrices into TEMATH and store them in the memory locations A, B, and C, do the following:

Select Calculators - Matrix Calculator... from the Work menu.
Double-click the Resize command in the matrix command list located in the lower left of the Matrix Calculator window.
Type 2, 2 into the command area located near the top of the Matrix Calculator window. Press the Enter key.
To enter the matrix A, type 3 Enter -5 Enter 23 Enter 0.

Warning: Here "Enter" means to press the Enter key and not the Return key!.

Click the store button and then the A button. This saves the matrix in memory location A for future use.
Click in the cell located in the first row, first column of the active matrix.
To enter the matrix B, type -12 Enter 20 Enter 2 Enter 3.
Click the store button and then the B button.
Double-click the Resize command.
Type 2, 3 into the command area located near the top of the Matrix Calculator window. Press the Enter key.
Click in the cell located in the first row, first column of the matrix.
To enter the matrix C, type 1 Enter 0 Enter 1 Enter -1 Enter 1 Enter 1.
Click the store button and then the C button.

In the middle of a calculation, the Matrix Calculator automatically recalls a matrix when a memory button is clicked. For example, to carry out a calculation involving scalar multiplication such as

4A - 3/2 B

you need only to do the following:

Click the recall button then click A.
Click the * button, type 4, click the enter button,
Click the - button, click the B button, click the * button, type 3/2, and click the enter button.

If you make a mistake entering an operation, for example, you click the "-" button instead of the "*" button, you can clear the pending operation by clicking the ce (Clear Entry) button. Also, notice that when you type a scalar or scalar expression into the command area, an OK button appears to the left of the command area. Clicking the OK button evaluates the scalar expression. If the value is the one that you want, click the enter button to complete the pending operation, otherwise, click the ce button to clear the pending operation and to start over.

Assuming A and B are square matrices, an expression involving the identity matrix I such as

5A - 4I + B

can be evaluated by doing the following:

Click the recall button then click A.
Click the * button, type 5, click the "enter" button,
Click the - button, click the I button,
Click the * button, type 4, click the "enter" button,
Click the + button, click the B button, and click the enter button.

To evaluate an expression involving matrix multiplication such as

ABC

Click the recall button then click A.
Click the * button, the B button, the * button, the C button, and the enter button.

To evaluate an expression involving matrix multiplication and parentheses such as

A(B - A) - I

Click the recall button then click B.
Click the - button, the A button, and the enter button.
Click the store button then click D.
You must compute B - A first and store it since matrix multiplication is not commutative.
Click the recall button then click A.
Click the * button, the D button, the - button, the I button, and the enter button.

To raise a matrix to a power, for example, A^5,

Click the recall button then click A.
Double-click the M^? command.
Type 5 into the command area and click the enterbutton.

Using the Matrix Commands

In the following, we will work with several of the matrix commands contained in the scrolling list located along the bottom left edge of the Matrix Calculator.

Last 20 matrix commands

To enter the 5 X 5 matrix M, where the element in the i-th row and j-th column is given by mij = i - j r (r is a random number between 0 and 1),

Double-click the Resize command, type 5, 5 into the command area, and press the Enter key.
Double-click the Fill command. Type the fill function m(i,j) = i - j rand into the command area and press the Enter key.
Click the store button and then the A button. This saves the matrix in memory location A for future use.

To find the Determinant of the matrix M,

Double-click the Det command.
Note that the value of the determinant is written into the command area of the Matrix Calculator window.

To find the matrix norm

Double-click the command.
The value of is written into the command area.

To find the eigenvalues of the matrix M,

Double-click the EigenVal command.
The eigenvalues are written into the matrix area of the Matrix Calculator window.

To find the inverse of the matrix M,

Click the recall button then click A.
Double-click the Inverse command.
The inverse matrix is written into the matrix area of the Matrix Calculator window.

To verify that M^(-1)M = I, where M^(-1) is the inverse matrix of M and I is the identity matrix,

Click the * button, the A button, and the enter button.

Using Elementary Row Operations

To find the row reduced echelon form of the matrix

the Matrix N

Double-click the Resize command, type 2, 2 into the command area, and press the Enter key.
Click in the cell located in the first row, first column of the matrix.
Type 2 enter 6 enter 3 enter -4.
Click the store button and then the A button. This saves the matrix in memory location A for backup use in case you make a mistake.
Move the mouse until the cursor is between the row label in the first row and the matrix (the cursor will change its shape to a small right-pointing arrow) and click the mouse button to highlight (select) the row.
Click the * button
Type 1/2 into the command area,
Press the Enter key.
Select the second row of the matrix.
Double-click the Element... command.
The template "R2 <- R2 + R? * ?" is written into the command area.
Select the first row of the matrix (note that R? is replaced with R1), type (-3),

You also could have simply edited the template to read "R2 <- R2 + R1 * (-3)". The parentheses around the -3 are necessary, otherwise, you would have two arithmetic operators next to each other *-.
Press the Enter key.
Select the second row of the matrix.
Click the * button, type 1/(-13) into the command area,
Press the Enter key.
Select the first row of the matrix.
Double-click the Element... command.
The template "R1 <- R1 + R? * ?" is written into the command area.
Select the second row of the matrix, type (-3),
Press the Enter key.