**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 [2] is false then display "Input matrix is not a square matrix!" and exit program

4. If step [2] 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 [5] is true for atleast one $A_{i,j}$ then display "The matrix is not a lower triangular matrix" and exit program.

7. If step [5] 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

# *************************************** # 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!