Python program to check if a given matrix is a lower triangular has been shown here. A square matrix is considered a lower triangular matrix if all the elements above the main diagonal are Zero. The elements in the main diagonal may or may not be 0.

Examples:

In the given examples, (i), (ii), (iv) and (v) are lower triangular matrices as every element above the principal diagonal is 0 in each of these matrices. The matrix shown in (iii) is not a lower triangular matrix as one element above the principal diagonal is non-zero. The dotted lines have been drawn over the principal diagonals.

## 1. Algorithm to Check If a Matrix Is a Lower Triangular Matrix

1. Take a matrix $A_{m\times n}$ as input

2. Check if $m=n$

3. If step  is false then display "Input matrix is not a square matrix!" and exit program

4. If step  is true, then

5. Check if $A_{i,j} \neq 0$ for each $i < j$ and $i \in [1, m]$ and $j \in [1, n]$

6. If step  is true for atleast one $A_{i,j}$ then display "The matrix is not a lower triangular matrix" and exit program.

7. If step  is false then display "The matrix is a lower triangular matrix" and exit program.

## 2. Pseudocode to Check If a Matrix Is a Lower Triangular Matrix

Input: A matrix $A_{m\times n}$

Output: A is lower triangular or not

1. Procedure lowerTriangularMatrix($A_{m\times n}$):

2. If $m == n$:

3. Repeat for each $i < j$ where $i \in [1, m]$ and $j \in [1, n]$

4. If $A_{i,j} \neq 0$:

5. Return Not a lower triangular matrix

6. Return A lower triangular matrix

7. Else:

8. Return It should be a square matrix

9. End Procedure

## 3. Time Complexity to Check If a Matrix Is a Lower Triangular Matrix

Time Complexity: O($mn$)

Where $m$ is the number of rows and $n$ is the number of columns in the matrices.

## 4. Python Program to Check If a Matrix Is a Lower Triangular Matrix

Code has been copied
# ***************************************
#        alphabetacoder.com
#  Python Program to check if a matrix
#   is a lower triangular matrix
# ***************************************

# declare and initialize array
A = [[0 for j in range(10)] for i in range(10)]
B = [[0 for j in range(10)] for i in range(10)]

# take input of the order of the matrix
m = int(input("Enter the number of rows of matrix = "))
n = int(input("Enter the number of columns of matrix = "))

# take input of the first matrix
print("Enter the elements of matrix of order", m, "x", n, "=")
for i in range(0, m):
for j in range(0, n):
A[i][j] = int(
input("Enter element of row {} and column {}: ".format(i + 1, j + 1))
)

# check if the matrix is a square matrix or not
# if it is square matrix, check if it is lower triangular or not
flag = 0
if m != n:
print("Input matrix is not a square matrix!")
else:
# check if A is lower triangular or not by
# finding an non-zero element above main diagonal
for i in range(0, m):
for j in range(i + 1, n):
if A[i][j] != 0:
# non-zero element found
flag = 1
break
# break the loop as non-zero element
# above main diagonal has been detected
if flag == 1:
break

# display the result
if flag == 0:
print("It is a lower triangular matrix!")
else:
print("It is not a lower triangular matrix!")


Output

Enter the number of rows of matrix = 3

Enter the number of columns of matrix = 3

Enter the elements of matrix of order 3 x 3 =

Enter element of row 1 and column 1: 1

Enter element of row 1 and column 2: 0

Enter element of row 1 and column 3: 0

Enter element of row 2 and column 1: 2

Enter element of row 2 and column 2: 3

Enter element of row 2 and column 3: 0

Enter element of row 3 and column 1: 5

Enter element of row 3 and column 2: 1

Enter element of row 3 and column 3: 6

It is a lower triangular matrix!