Vectors
and Matrices
To write a vector
of the form 
, type \doubleA
followed with the space bar to
get the optional character font for A and use the Matrix feature.
, type \doubleA
followed with the space bar to
get the optional character font for A and use the Matrix feature.
Similarly, the vector 
requires \doubleB
for the font for B. After entry of a
vector or matrix, select Calculate to
place the parentheses.
requires \doubleB
for the font for B. After entry of a
vector or matrix, select Calculate to
place the parentheses.
Example 1: Add two vectors.
The command Calculate gives the resultant vector,
Unfortunately, the input 
does not give the answer. The software does not assign the vectors to
what was previously defined.
does not give the answer. The software does not assign the vectors to
what was previously defined.
Example 2: Find the magnitude of a vector.
Enter a vector as a string with { }. Separate components of the vector with a
comma. Use the command Calculate to find the answer:
Example 3: Find the inner product.
Separate vectors with a comma. Use the command Calculate to yield:
Example 4: Find the cross product where 
and 
and
Enter the vectors as strings. Separate vectors with a comma. Use the command cross. The strings are 
. The input is:
. The input is:
Select
Calculate to give the vector that is the cross product.
Example 5: Calculate the determinant of a 3 x 3 matrix.
Highlight, right-click,
and select Calculate Determinant to
find the answer of 
for our example.
for our example. 
Using the other options, find the trace
of the matrix is 5, the inverse
matrix is 
and
the transpose of the matrix is 
.
.
Example 6: Given a 3 x 3 matrix and its inverse, show
the product gives the identity matrix.
No symbol is needed between the
matrices. The input window needs to be
one blue window for both matrices as shown below:
The command Calculate gives the answer of the identity matrix below:
Example 7: Find the identity 8 x 8 matrix.
The command identityMatrix(n) gives the identity n x n amatrix where n is
from one to fifteen. The Calculate
option gives our answer:
Example 8: Find the Boolean product of
.
.
There are a few ways to do this. For one approach, enter the 3 x 3 matrix 
and do not press Calculate. The insertion of
a pasted object with spaces or parentheses often gives an error message. Without pressing Calculate, the matrix, will
contain no parentheses. Cut and paste
this matrix and place as the ‘base’.
Insert the three as a superscript using this feature:
and do not press Calculate. The insertion of
a pasted object with spaces or parentheses often gives an error message. Without pressing Calculate, the matrix, will
contain no parentheses. Cut and paste
this matrix and place as the ‘base’.
Insert the three as a superscript using this feature:
The command Calculate from the drop-down menu gives
the desired result of:
The superscript may be toggled to be 7,
and the new Boolean product can be found to be:
=
Another
approach is to enter the matrix and superscript directly without cutting and
pasting to give: 
If there is a desire for the parentheses
to be displayed, enter the matrix and then press Calcluate to give:
With the cursor inside the blue box,
press ^3 followed by a space bar.
This will move the three to the desired
location and give:
Press Calculate
to simplify.
Example 9: Row reduce the 2 x 2 matrix.
Use the command reduce followed by the matrix.
Calculate
will give the answer:
Example 10: Reduce a 3 x 3 matrix to row reduced echelon
form (Lipschutz, 1968,p. 53).
Enter the 3 x 3 matrix using Matrix:
The input will look like this:
Press Calculate
to obtain the parentheses.
Right-click to bring open a blue
box. Place the cursor within the blue
box and in front of the parentheses.
Type reduce.
The command Calculate will yield:
Example
11: Reduce the matrix to row reduced echelon form
(Lipschutz, 1968, p.52).
There is a problem entering this size of matrix from the
menu since it is larger than a 3 x 3 matrix.
Enter each row as a string. Separate each string with a comma. Recall, a string requires the notation, { and
}. With Insert New Equation, type:
Calculate will
place { } around the input.
With the cursor in the blue box, type matrix
The input line is:
Right-click and select Calculate gives our matrix:
With the cursor within the blue box and in front of the
first parenthesis, type reduce.
Right click and press Calculate
to execute:
The answer is:
Example
12: Find the inverse of a matrix.
If the matrix is a 2 x 2 or 3 x 3 matrix, the pull-down menu will contain the option, Invert Matrix.
However, for larger matrices
the option will appear with the command, matrix, in the Insert New Equation line.
Enter each row of
the matrix as a string using { and }.
Separate the elements with a comma.
Consider the matrix
To enter the matrix,
type:
Place the cursor in the
blue box and in front of the desired matrix.
Type matrix.
With the insertion of matrix, the pop-up box changes and the
desired option of Invert Matrix appears.
The answer is:
Example 13: Solve a system of linear equations.
Type each equation in a separate Insert New Equation line. If the last equation had no term of 7y, then a place-holder of 0y would have been needed. Now, highlight all three and right-click. The
menu is:
Select Solve for x, z, y. The answer is:
An alternate way is to set-up an
augmented matrix. Insert:
Right-click on Calculate to give:
Type reduce.
Right-click to bring up the menu and the
option, Calculate. The output is:
Interpret the answer to be:
References
Lipschutz, Seymour. , Schaum’s
Outline Series Theory and Problems of Linear Algebra, McGraw-Hill, New
York, 1968.
Zill, D. and Cullen, M. Advanced Engineering Mathematics,
third edition, Jones and Bartlett, Sudbury, Massachusetts, 2006).
Linear
Algebra
Example 1: Find the eigenvalues of the matrix of
coefficients.
There are some common symbols in Basic Math.
An alternate way to reach them is to type
a back-slash (\) followed by what you
would like. Pressing the space bar will insert the object. For example, \pi followed by the space bar will insert 
. Lambda is not on our
list. We can insert lambda with the
following input:
. Lambda is not on our
list. We can insert lambda with the
following input:
To execute, use the space bar to toggle
this to 
.
.
Insert the desired matrix using the
ribbon icon Matrix.
Selecting Calculate from the menu will place the parentheses.
Right-click to bring up the following
menu:
Select Calculate Determinant to yield:
Cut and paste this expression and set
equal to zero.
The menu will allow you to solve for
lambda.
The answer is:
Example 2: Find the characteristic equation of the
matrix A and the eigenvalues of the matrix of coefficients.
The input must be within one blue frame
as shown below:
No operation is needed between lambda and
the identity matrix. A short-cut to
insert the identity matrix is the command IdentityMatrix
(n) where n is the number of rows
of the square matrix. The option, Calculate,
will insert the parentheses and the 3
x 3 identity matrix. Simplify
from the pull-down menu will give:
Right-click and press Calculate Determinant. Cut and paste the
result into a new equation line.
Set equal to zero to form an equation, and
select Solve for .
.
The answer is:
References
Lipschutz, Seymour. Schaum’s
Outline Series Theory and Problems of Linear Algebra, McGraw-Hill, New
York, 1968.
Zill, D. and Cullen, M. Advanced
Engineering Mathematics, third edition, Jones and Bartlett, Sudbury,
Massachusetts, 2006).
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