C++ program to check if a given matrix is an upper triangular has been shown here. A square matrix is considered an upper triangular matrix if all the elements below 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 upper triangular matrices as every element below the principal diagonal is 0 in each of these matrices. The matrix shown in (iii) is not an upper 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 an Upper 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 an upper triangular matrix" and exit program.

7. If step  is false then display "The matrix is an upper triangular matrix" and exit program.

## 2. Pseudocode to Check If a Matrix Is an Upper Triangular Matrix

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

Output: A is upper triangular or not

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

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 an upper triangular matrix

6. Return An upper triangular matrix

7. Else:

8. Return It should be a square matrix

9. End Procedure

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

Time Complexity: O($mn$)

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

## 4. C++ Program to Check If a Matrix Is an Upper Triangular Matrix

Code has been copied
/**********************************
alphabetacoder.com
C++ Program to check if a matrix
is an upper triangular matrix
***********************************/
#include <iostream>

using namespace std;

int main() {
// declare variables
int m, n, i, j, flag = 0;
int A = {0};

//take input of the order of the matrix
cout << "Enter the number of rows and columns of matrix = ";
cin >> m >> n;

//take input of the first matrix
cout << "Enter the elements of matrix of order " << m << " x " << n << " = " << endl;
for (i = 0; i < m; i++)
for (j = 0; j < n; j++)
cin >> A[i][j];

// check if the matrix is a square matrix or not
// if it is square matrix, check if it is upper triangular or not
if (m != n)
cout << "\nInput matrix is not a square matrix!" << endl;
else {
// check if A is upper triangular or not by
// finding an non-zero element below main diagonal
for (i = 0; i < m; i++) {
for (j = 0; j < i; j++) {
if (A[i][j] != 0) {
// non-zero element found
flag = 1;
break;
}
}
// break the loop as non-zero element
// below main diagonal has been detected
if (flag == 1)
break;
}

// display the result
if (flag == 0)
cout << "\nIt is an upper triangular matrix!";
else
cout << "\nIt is not an upper triangular matrix!";
}

return 0;
}


Output

Enter the number of rows and columns of matrix = 4 4

Enter the elements of matrix of order 4 x 4 =

0 5 7 8

0 1 2 0

0 0 0 3

0 0 0 6

It is an upper triangular matrix!