After reading this chapter, you should be able to:
1. know what unary ope rations means ,
2. find the transpose of a square matrix and it’s relationship to symmetric
matrices,
3. find the trace of a matrix, and
4. find the de terminant of a matrix by the co factor method .
What is the transpose of a matrix?
Let 
be a 
matrix. Then 
is the transpose of
the if 
for all i and j
That is,
the row and the
column element of
is the
row and column
element of . Note,
would be a
matrix. The transpose of
is denoted by .
Example 1
Find the transpose of
Solution
The transpose of is
Note, the transpose of a row vector is a column vector and
the transpose of a column vector
is a row vector.
Also, note that the transpose of a transpose of a matrix is the matrix itself,
that is,
What is a symmetric matrix?
A square matrix with
real elements where
for
and 
is
called a symmetric matrix. This is same as, if
, then 
is
a symmetric matrix.
Example 2
Give an example of a symmetric matrix.
Solution
is a symmetric matrix as
and 
.
What is a skew-symmetric matrix?
A 
matrix is skew symmetric if
for 
and
This is same as
Example 3
Give an example of a skew-symmetric matrix.
Solution
is skew-symmetric as
Since 
only if 
, all
the diagonal elements of a skew-symmetric matrix have to be zero.
What is the trace of a matrix?
The trace of a matrix
is the sum of the
diagonal entries of ,
that is,
Example 4
Find the trace of
Solution
Example 5
The sales of tires are given by make (rows) and quarters (columns) for B lowout
r ’us store
location A, as shown below.
where the rows re present the sale of Tirestone, Michigan
and Copper tires, and the columns
represent the quarter number 1, 2, 3, 4.
Find the total yearly revenue of store A if the prices of tires vary by quarters
as follows.
where the rows represent the cost of each tire made by
Tirestone, Michigan and Copper, and
the columns represent the quarter numbers.
Solution
To find the total tire sales of store A for the whole year, we need to find the
sales of each
brand of tire for the whole year and then add to find the total sales. To do so,
we need to
rewrite the price matrix so that the quarters are in rows and the brand names
are in the
columns, that is, find the transpose of
.
Recognize now that if we find
, we get
The diagonal elements give the sales of each brand of tire
for the whole year,
that is
(Tirestone sales)
(Michigan sales)
(Cooper sales)
The total yearly sales of all three brands of tires are
and this is the trace of the matrix
.
Define the determinant of a matrix.
The determinant of a square matrix is a single unique real number corresponding
to a matrix.
For a matrix ,
determinant is denoted by 
or
. So do not use
and
inter changeably .
For a 2×2 matrix
How does one calculate the determinant of any square
matrix?
Let be
matrix. The minor of entry
is denoted by
and is defined as the
determinant of the 
submatrix of
, where the submatrix
is obtained by
deleting the row and
column of the matrix
. The determinant is
then given by
or
coupled with that 
for
a 1×1 matrix , as we
can always reduce the determinant
of a matrix to determinants of 1×1matrices. The number
is called the cofactor of
and is denoted by
. The above equation for the determinant can
then be written as
or
The only reason why determinants are not generally
calculated using this method is that it
becomes computationally intensive. For a
matrix, it
requires arithmetic operations
proportional to n !.
Example 6
Find the determinant of
Solution
Method 1:
Let i=1 in the formula
Also for i=1
Method 2:
In terms of cofactors for j=2
Is there a relationship between det(AB), and det(A) and
det(B)?
Yes, if and
are square matrices
of same size, then
Are there some other theorems that are important in finding the determinant
of a
square matrix?
Theorem 1: If a row or a column in a
matrix
is zero, then .
Theorem 2: Let
be a matrix. If a row
is proportional to another row, then
Theorem 3: Let
be a matrix. If a
column is proportional to another column, then
Theorem 4: Let
be a matrix. If a
column or row is multiplied by to result in
matrix k, then 
Theorem 5: Let
be a upper or lower
triangular matrix, then 
Example 7
What is the determinant of
Solution
Since one of the columns (first column in the above example) of
is a zero,
Example 8
What is the determinant of
Solution
is zero because
the fourth column
is 2 times the first column
Example 9
If the determinant of
is -84, then what is the determinant of
Solution
Since the second column of
is 2.1 times the
second column of
Example 10
Given the determinant of
is -84, what is the determinant of
Solution
Since is simply
obtained by subtracting the second row of
by 2.56 times the
first
row of ,
Example 11
What is the determinant of
Solution
Since is an upper
triangular matrix
Key Terms
Transpose
Symmetric Matrix
Skew-Symmetric Matrix
Trace of Matrix
Determinant
