C++ program to find trace of a given matrix has been shown here. Suppose A is a matrix, then trace of matrix A is denoted as tr(A) which is sum of elements on main diagonal. Only a square matrix can have a trace.

Examples:

$A = \begin{bmatrix} 1 & 5 & 6\\ 0 & 4 & 3\\ 2 & 7 & 8 \end{bmatrix}$

So, the trace of matrix A = 1 + 4 + 8 = 13

## 1. Algorithm to Find Trace of a 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. Set $sum = 0$

6. For each $i = j$, successively add $A_{ij}$ to $sum$, where $i \in [1, m]$ and $j \in [1, n]$

7. Declare $sum$ as the trace of the matrix and exit program.

## 2. Pseudocode to Find Trace of a Matrix

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

Output: Trace of matrix $A_{m\times n}$

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

2. If $m == n$:

3. $sum \leftarrow 0$

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

5. $sum \leftarrow sum + A_{i,j}$

6. Return $sum$

7. Else:

8. Return It should be a square matrix

9. End Procedure

## 3. Time Complexity to Find Trace of a Matrix

Time Complexity: O($m$)

Where $m$ is the number of rows and columns in the given square matrix.

## 4. C++ Program to Find Trace of a Matrix

Code has been copied
/*************************************
alphabetacoder.com
C++ Program to find trace of a matrix
**************************************/
#include <iostream>

using namespace std;

int main() {
// declare variables
int m, n, i, j, sum;
int A[10][10] = {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 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, find the trace
if (m != n)
cout << "\nInput matrix is not a square matrix!" << endl;
else {
// initialize
sum = 0;
// find trace of matrix
for (i = 0; i < m; i++) {
sum += A[i][i];
}
cout << "\nTrace of given matrix: " << sum;
}

return 0;
}


Output

Enter the number of rows and columns of matrix = 3 3

Enter the elements of matrix of order 3 x 3 =

5 6 4

0 2 1

1 2 0

Trace of given matrix: 7