The transpose of a matrix is obtained by converting its rows into columns and its columns into rows. In other words, for a matrix A, the element at A[i][j] moves to B[j][i] in the transpose.
- For a matrix of size M × N, its transpose has size N × M.
- A square matrix can also be transposed in-place without using an extra matrix.
Example:
Approaches to Find the Transpose of a Matrix
The transpose of a matrix can be found using the following approaches:
1. Transpose of a Square Matrix Using an Extra Matrix
For a square matrix, create another matrix B of the same size. For every element A[i][j], store it at B[j][i].
- Create a result matrix B of the same size as A.
- Traverse every element of A.
- Store A[i][j] in B[j][i].
- Print the resulting matrix.
#include <iostream>
using namespace std;
#define N 4
// Stores the transpose of A in B
void transpose(int A[][N], int B[][N])
{
for (int i = 0; i < N; i++) {
for (int j = 0; j < N; j++) {
B[j][i] = A[i][j];
}
}
}
int main()
{
int A[N][N] = {
{1, 1, 1, 1},
{2, 2, 2, 2},
{3, 3, 3, 3},
{4, 4, 4, 4}
};
int B[N][N];
transpose(A, B);
cout << "Transpose of the matrix:\n";
for (int i = 0; i < N; i++) {
for (int j = 0; j < N; j++) {
cout << B[i][j] << " ";
}
cout << '\n';
}
return 0;
}
Output
Transpose of the matrix: 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
Explanation: The function traverses the original matrix and swaps the row and column indices while storing each element in B[j][i]. Thus, the rows of A become the columns of B.
2. Transpose of a Rectangular Matrix Using an Extra Matrix
For a rectangular matrix of size M × N, the transpose has dimensions N × M. Therefore, the result matrix must have N rows and M columns.
- Create a result matrix B of size N × M.
- Traverse the original M × N matrix.
- Store A[i][j] in B[j][i].
- Print the resulting N × M matrix.
#include <iostream>
using namespace std;
#define M 3
#define N 4
// Stores the transpose of A in B
void transpose(int A[][N], int B[][M])
{
for (int i = 0; i < M; i++) {
for (int j = 0; j < N; j++) {
B[j][i] = A[i][j];
}
}
}
int main()
{
int A[M][N] = {
{1, 1, 1, 1},
{2, 2, 2, 2},
{3, 3, 3, 3}
};
// Transpose has N rows and M columns
int B[N][M];
transpose(A, B);
cout << "Transpose of the matrix:\n";
for (int i = 0; i < N; i++) {
for (int j = 0; j < M; j++) {
cout << B[i][j] << " ";
}
cout << '\n';
}
return 0;
}
Output
Transpose of the matrix: 1 2 3 1 2 3 1 2 3 1 2 3
Explanation: The original matrix has M rows and N columns, while its transpose has N rows and M columns. The program stores every A[i][j] at B[j][i], which converts rows into columns.
3. In-Place Transpose of a Square Matrix
A square matrix can be transposed without creating another matrix. The idea is to swap elements across the main diagonal. For every pair where j > i, swap A[i][j] with A[j][i]. Elements on the main diagonal remain unchanged.
- Traverse the upper triangular part of the matrix.
- For every element A[i][j] where j > i, swap it with A[j][i].
- The matrix is now transposed in-place.
#include <utility>
using namespace std;
#define N 4
// Transposes the matrix in-place
void transpose(int A[][N])
{
for (int i = 0; i < N; i++) {
for (int j = i + 1; j < N; j++) {
swap(A[i][j], A[j][i]);
}
}
}
int main()
{
int A[N][N] = {
{1, 1, 1, 1},
{2, 2, 2, 2},
{3, 3, 3, 3},
{4, 4, 4, 4}
};
transpose(A);
cout << "Transpose of the matrix:\n";
for (int i = 0; i < N; i++) {
for (int j = 0; j < N; j++) {
cout << A[i][j] << " ";
}
cout << '\n';
}
return 0;
}
Output
Transpose of the matrix: 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4
Explanation: Instead of creating a second matrix, the program directly swaps elements on opposite sides of the main diagonal. Since each pair is swapped only once, the matrix is transposed without using additional storage.