Diagonal, Triangular, and Symmetrical Matrices
There are many types of matrices with special properties that include transposes,
inverses, and matrix operations. These matrices can either be diagonal, triangular, or
symmetrical.
Triangular Matrices:
Triangular matrices can fall into one of two categories—lower triangular or upper
triangular.
Lower Triangular Matrices:
A lower triangular matrix is a matrix where there are zeros above the main
diagonal.
− 3 0

 2 4
Examples: 
1 0 0 
2 1 0


3 2 1 
− 3 0
 2 −2

1
1

1
1
0 
0 0 
1 0 

1 − 4
0
1. Inverses of Lower Triangular Matrices
The inverse of a lower triangular matrix is also lower triangular.
1 0 0 


Example: Find the inverse of 2 1 0 , if it exists.


3 2 1 
a. Attach the identity matrix to the given matrix.
1 0 0 1 0 0 


2 1 0 0 1 0
3 2 1 0 0 1
b. Reduce A to In using the Gauss-Jordan Elimination method.
1 0 0 1
1 0 0 1 0 0 R 2− 2 R1 1 0 0 1 0 0
0 0
 R 3− 2 R 2 


 R 3−3 R1 
→0 1 0 − 2 1 0
2 1 0 0 1 0  → 0 1 0 − 2 1 0  
3 2 1 0 0 1
0 2 1 − 3 0 1
0 0 1 − 3 − 2 1 
c. Check the answer by multiplying AA-1. This should yield the identity matrix In.
If it doesn’t, go back and check your work.
0 0 1 0 0
1 0 0   1
 2 1 0   − 2 1 0  = 0 1 0 


 

3 2 1   − 3 − 2 1  0 0 1 
2. Multiplying Lower Triangular Matrices
The product of two or more lower triangular matrices is also lower
triangular.
− 3 0 − 3 0
 
.
 2 4  2 4
Example: Compute 
Multiplying these two lower triangular matrices together yields
 − 3 0   − 3 0  9 0 
 2 4  2 4 = 2 16
 


 
Hence, the product is also lower triangular.
3. Transpose of Lower Triangular Matrices
The transpose of a lower triangular matrix is upper triangular.
1 0 0 


Example: Find the transpose of 2 1 0 .


3 2 1 
1 0 0 


Transposing the matrix yields 2 1 0 , which is an upper triangular matrix.


3 2 1 
Upper Triangular Matrices:
An upper triangular matrix is a matrix where there are zeros below the main
diagonal.
1 2 3
0 1 2 


0 0 1 
 − 3 2

 0 4
Examples: 
− 3 2
 0 −2

0
0

0
0
1 
1 1 
0 1 

0 − 4
1
1. Inverses of Upper Triangular Matrices
The inverse of an upper triangular matrix is also upper triangular.
 − 3 2
 , if it exists.
 0 4
Example: Find the inverse of 
Using the formula to find the inverse of a 2 x 2 matrix gives
A=
1  4 − 2
1 − 1 / 3 1 / 6
=


1 / 4
− 12 0 − 3 − 12  0
2. Multiplying Upper Triangular Matrices
The product of two or more upper triangular matrices is also lower
triangular.
1 2 3 1 2 3



Example: Compute 0 1 2 0 1 2 .



0 0 1  0 0 1 
Multiplying these two upper triangular matrices together yields
1 2 3 1 2 3 1 4 10
0 1 2  0 1 2  = 0 1 4 


 

0 0 1  0 0 1  0 0 1 
Hence, the product is also upper triangular.
4. Transpose of Upper Triangular Matrices
The transpose of an upper triangular matrix is lower triangular.
− 3
0
Exmaple: Find the transpose of 
0

0
− 3 0 0
 2 −2 0
Transposing the matrix yields 
1
1 0

1 1
1
1 
− 2 1 1 
.
0 0 1 

0 0 − 4
0 
0 
, which is a lower triangular
0 

− 4
2
1
matrix.
Symmetric Matrices:
A symmetric matrix is a matrix which its transpose is equal to itself, ie AT = A.
1 2 3
2 1 2


3 2 1 
 − 3 2

 2 4
Examples: 
− 3 2
 2 −2

1
1

1
1
1. Inverse of Symmetric Matrices
The inverse of a symmetric matrix is also symmetric.
 − 3 2
 , if it exists.
 2 4
Example: Find the inverse of 
Using the formula to find the inverse of a 2 x 2 matrix gives
A −1 =
1  4 − 2
1 − 1 / 4 1 / 8 
=


− 16 − 2 − 3 − 16  2 / 8 − 3 / 16
2. Multiplying Symmetric Matrices
The product of two symmetric matrices is also symmetric.
1 
1 1 
1 1 

1 − 4
1
 − 3 2  − 3 2
 
.
 2 4  2 4
Example: Compute 
The product of these two symmetric matrices is given by
− 3 2 − 3 2 13 2 
 2 4  2 4 =  2 20
 

 

3. Transpose of Symmetric Matrix
The transpose of any symmetric matrix is the original matrix itself.
1 2 3


Example: Find the transpose of 2 1 2 .


3 2 1 
1 2 3


Transposing the matrix yields 2 1 2 , which is still symmetric and its


3 2 1 
original form, ie, AT = A.
Diagonal Matrices:
A diagonal matrix is a matrix where it is upper triangular, lower triangular, and
symmetric.
− 3 0

 0 4
Examples: 
1 0 0 
0 2 0 


0 0 − 3
− 3 0
 0 −2

0
0

0
0
0 
0 0 
0 0 

0 − 4
0
1. Inverses of Diagonal Matrices
The inverse of a diagonal matrix is just the reciprocal of the elements on the
diagonal. If one of the diagonal entries is zero, then the inverse doesn’t exist.
Finding the inverse of the above matrices gives the following:
− 1 / 3 0 
Examples: 
1 / 4
 0
0 
1 0
0 1 / 2
0 

0 0 − 1 / 3
Inverse doesn’t exist.
2. Multiplying Diagonal Matrices
To multiply a diagonal matrix on the left, just multiply the corresponding
diagonal entry to each row of the matrix. To multiply a diagonal matrix on the
right, just multiply the corresponding diagonal entry to each column of the
matrix.
3 0 0 


Examples: 0 2 0


0 0 − 2
1 2 3 
4 5 6


7 8 9 
(3)2
(3)3   3
6
9 
1 2 3  (3)1
4 5 6 =  (2)4 (2)5 (2)6  =  8
10
12 

 
 
7 8 9  (−2)7 (−2)8 (−2)9 − 14 − 16 − 18
3 0 0   (3)1 (2)2 (−2)3  3 4 − 6 
0 2 0  = (3)4 (2)5 (−2)6 = 12 10 − 12

 
 

0 0 − 2 (3)7 (2)8 (−2)9 21 16 − 18
3. Transpose of Diagonal Matrices
The transpose of any diagonal matrix is still itself.