Microsoft Word Free Math Add-In
Linear Algebra
Vectors and Matrices
1
To write a vector of the form 𝔸 = (2) , type \doubleA followed with the space bar to get the optional
3
character font for A and use the Matrix feature.
4
Similarly, the vector 𝔹 = (−5) requires \doubleB for the font for B. After entry of a vector or matrix,
3
select Calculate to place the parentheses.
Example 1: Add two vectors.
1
4
(2) + (−5)
3
3
The command Calculate gives the resultant vector,
5
(−3)
6
©2009 Nord
Page 1
Microsoft Word Free Math Add-In
Unfortunately, the input 𝔸 + 𝔹 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.
𝑚𝑎𝑔𝑛𝑖𝑡𝑢𝑑𝑒({3, −2,1})
Enter a vector as a string with { }. Separate components of the vector with a comma. Use the command
Calculate to find the answer:
√14
Example 3: Find the inner product.
𝑖𝑛𝑛𝑒𝑟({3, −2,1}, {5,0,2})
Separate vectors with a comma. Use the command Calculate to yield:
17
Example 4: Find the cross product where 𝔸 =< −2, 4, 5 > and 𝔹 =< 3, 2, 1 >.
Enter the vectors as strings. Separate vectors with a comma. Use the command cross. The strings are
{−2,4,5} and {3,2,1}. The input is:
𝑐𝑟𝑜𝑠𝑠({−2,4,5}, {3,2,1})
Select Calculate to give the vector that is the cross product.
{−6, 17, −16}
Example 5: Calculate the determinant of a 3 x 3 matrix.
1
1 2
( 2 −2 4)
−3 3 6
Highlight, right-click, and select Calculate Determinant to find the answer of −48 for our example.
©2009 Nord
Page 2
Microsoft Word Free Math Add-In
1
2
1
2
Using the other options, find the trace of the matrix is 5, the inverse matrix is
(
1
and the transpose of the matrix is (1
2
0
0
1
1
−6
−4
0
1
8
1
12
)
2 −3
−2 3 ) .
4
6
Example 6: Given a 3 x 3 matrix and its inverse, show the product gives the identity matrix.
1
1
0 −
2
6
1
1 2
1
1
−
0 ( 2 −2 4)
2
4
−3 3 6
1
1
(0 8
12 )
No symbol is needed between the matrices. The input window needs to be one blue window for both
matrices as shown below:
©2009 Nord
Page 3
Microsoft Word Free Math Add-In
The command Calculate gives the answer of the identity matrix below:
1
(0
0
0 0
1 0)
0 1
Example 7: Find the identity 8 x 8 matrix.
𝑖𝑑𝑒𝑛𝑡𝑖𝑡𝑦𝑀𝑎𝑡𝑟𝑖𝑥 (8)
The command identityMatrix(n) gives the identity n x n amatrix where n is from one to fifteen. The
Calculate option gives our answer:
1
0
0
0
0
0
0
(0
0
1
0
0
0
0
0
0
0
Example 8: Find the Boolean product of (1
1
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1
0
0
0
0
0
0
0
0
1)
0 1 3
0 0) .
1 0
0 0 1
There are a few ways to do this. For one approach, enter the 3 x 3 matrix 1 0 0 and do not press
1 1 0
Calculate. The insertion of a pasted object with spaces or parentheses often gives an error message.
0 0 1
Without pressing Calculate, the matrix, 1 0 0 will contain no parentheses. Cut and paste this
1 1 0
matrix and place as the ‘base’. Insert the three as a superscript using this feature:
©2009 Nord
Page 4
Microsoft Word Free Math Add-In
The command Calculate from the drop-down menu gives the desired result of:
1
(1
1
0 1
1 0)
1 1
The superscript may be toggled to be 7, and the new Boolean product can be found to be:
0 0
1 0
1 1
17
3 2
=
(
0
2 1
0
4 2
2
2)
3
Another approach is to enter the matrix and superscript directly without cutting and pasting to give:
0 0
1 0
1 1
13
0
0
If there is a desire for the parentheses to be displayed, enter the matrix and then press Calcluate to give:
0
(1
1
0 1
0 0)
1 0
With the cursor inside the blue box, press ^3 followed by a space bar.
©2009 Nord
Page 5
Microsoft Word Free Math Add-In
This will move the three to the desired location and give:
1 0
(1 0
1 1
1 3
0)
0
Press Calculate to simplify.
Example 9: Row reduce the 2 x 2 matrix.
3
(
6
1
)
2
Use the command reduce followed by the matrix.
𝑟𝑒𝑑𝑢𝑐𝑒
3 1
6 2
Calculate will give the answer:
(1
0
1
3)
0
Example 10: Reduce a 3 x 3 matrix to row reduced echelon form (Lipschutz, 1968,p. 53).
6 3 −4
(−4 1 −6)
1 2 −5
Enter the 3 x 3 matrix using Matrix:
©2009 Nord
Page 6
Microsoft Word Free Math Add-In
The input will look like this:
6 3 −4
−4 1 −6
1 2 −5
Press Calculate to obtain the parentheses.
6 3 −4
(−4 1 −6)
1 2 −5
Right-click to bring open a blue box. Place the cursor within the blue box and in front of the
parentheses.
Type reduce.
6 3
𝑟𝑒𝑑𝑢𝑐𝑒 (−4 1
1 2
−4
−6)
−5
The command Calculate will yield:
©2009 Nord
Page 7
Microsoft Word Free Math Add-In
1 0
0 1
(0 0
7
9
26
−
9
0 )
Example 11: Reduce the matrix to row reduced echelon form (Lipschutz, 1968, p.52).
1 −2 3 −1
(2 −1 2 2 )
3 1 2 3
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:
{1, −2,3, −1}, {2, −1,2,2}, {3,1,2,3}
Calculate will place { } around the input.
{{1, −2, 3, −1}, {2, −1, 2, 2}, {3, 1, 2, 3}}
With the cursor in the blue box, type matrix
Type in here
matrix.
The input line is:
𝑚𝑎𝑡𝑟𝑖𝑥{{1, −2, 3, −1}, {2, −1, 2, 2}, {3, 1, 2, 3}}
©2009 Nord
Page 8
Microsoft Word Free Math Add-In
Right-click and select Calculate gives our matrix:
1 −2 3 −1
(2 −1 2 2 )
3 1 2 3
With the cursor within the blue box and in front of the first parenthesis, type reduce.
Right click and press Calculate to execute:
1
𝑟𝑒𝑑𝑢𝑐𝑒 (2
3
The answer is:
1 0
0 1
(0 0
Example 12: Find the inverse of a matrix.
−2 3 −1
−1 2 2 )
1 2 3
15
7
4
0 −
7
10
1 − )
7
0
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,
2
5 −3 −2
−2 −3 2 −5
(
)
1
3 −2 2
−1 −6 4
3
To enter the matrix, type:
{{2,5, −3, −2}, {−2, −3,2, −5}, {1,3, −2,2}, {−1, −6,4,3}}
©2009 Nord
Page 9
Microsoft Word Free Math Add-In
Place the cursor in the blue box and in front of the desired matrix.
Type matrix.
𝑚𝑎𝑡𝑟𝑖𝑥{{2, 5, −3, −2}, {−2, −3, 2, −5}, {1, 3, −2, 2}, {−1, −6, 4, 3}}
With the insertion of matrix, the pop-up box changes and the desired option of Invert Matrix
appears.
The answer is:
7
4
17
4
23
4
1
4
0 −
2
3
(0
©2009 Nord
13
4
31
4
41
4
3
4
−
3
4
13
4
19
4
1
4 )
−
Page 10
Microsoft Word Free Math Add-In
Example 13: Solve a system of linear equations.
2𝑥 + 6𝑦 − 𝑧 = 7
𝑥 + 2𝑦 − 𝑧 = −1
5𝑥 + 7𝑦 − 4𝑧 = 9
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:
𝑥 = 12
{ 𝑧 = 11
𝑦 = −1
An alternate way is to set-up an augmented matrix. Insert:
𝑚𝑎𝑡𝑟𝑖𝑥{{2, 6, −1, 7}, {1, 2, −1, −1}, {5, 7, −4, 9}}
Right-click on Calculate to give:
2 6
(1 2
5 7
©2009 Nord
−1 7
−1 −1)
−4 9
Page 11
Microsoft Word Free Math Add-In
Place cursor
in blue box
at the front.
Type reduce.
2
𝑟𝑒𝑑𝑢𝑐𝑒 (1
5
6 −1 7
2 −1 −1)
7 −4 9
Right-click to bring up the menu and the option, Calculate. The output is:
1 0
(0 1
0 0
0 12
0 −1)
1 11
Interpret the answer to be:
𝑥 = 12
{ 𝑦 = −1
𝑧 = 11
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).
©2009 Nord
Page 12
Microsoft Word Free Math Add-In
Linear Algebra
Eigenvalues
Example 1: Find the eigenvalues of the matrix of coefficients.
1−𝜆
(
3
4
)
2−𝜆
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 𝜋. Lambda is not
on our list. We can insert lambda with the following input:
\𝑙𝑎𝑚𝑏𝑑𝑎
To execute, use the space bar to toggle this to 𝜆.
Insert the desired matrix using the ribbon icon Matrix.
©2009 Nord
Page 13
Microsoft Word Free Math Add-In
Selecting Calculate from the menu will place the parentheses.
1−𝜆
(
3
4
)
2−𝜆
Right-click to bring up the following menu:
Select Calculate Determinant to yield:
𝜆2 − 3 𝜆 − 10
©2009 Nord
Page 14
Microsoft Word Free Math Add-In
Cut and paste this expression and set equal to zero.
𝜆2 − 3 𝜆 − 10 = 0
The menu will allow you to solve for lambda.
The answer is:
𝜆=5
𝜆 = −2
Example 2: Find the characteristic equation of the matrix A and the eigenvalues of the matrix of
coefficients.
1 0
1 5 −9
(3 2 4 ) − 𝜆 (0 1
0 0
4 7 −2
0
0)
1
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:
©2009 Nord
Page 15
Microsoft Word Free Math Add-In
−𝜆 + 1
5
−9
( 3
−𝜆 + 2
4 )
4
7
−𝜆 − 2
Right-click and press Calculate Determinant. Cut and paste the result into a new equation line.
−𝜆3 + 𝜆2 + 11 𝜆 − 39
Set equal to zero to form an equation, and select Solve for 𝜆.
−𝜆3 + 𝜆2 + 11 𝜆 − 39 = 0
The answer is:
𝜆 ≈ −4.0654239158404
𝜆 ≈ 2.5327119579202 − 1.7828251932075𝑖
𝜆 ≈ 2.5327119579202 + 1.7828251932075𝑖
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).
©2009 Nord
Page 16