Transposition of a Matrix #
The transpose of a matrix is obtained by swapping rows and columns.
If
\[ A = [a_{ij}] \]then the transpose of ( A ), denoted ( A^T ), is:
\[ A^T = [a_{ji}] \]Rules of Matrix Transposition #
1. Transpose of a Transpose #
\[ (A^T)^T = A \]Intuition
Swapping rows and columns twice returns the original matrix.
2. Transpose of a Sum #
\[ (A + B)^T = A^T + B^T \]Condition
Matrices ( A ) and ( B ) must have the same dimensions.
Intuition
Transposition distributes over addition.
3. Transpose of a Scalar Multiple #
\[ (cA)^T = cA^T \]Intuition
Scaling a matrix does not affect how rows and columns are swapped.
4. Transpose of a Product #
\[ (AB)^T = B^T A^T \]Important
The order of multiplication reverses.
Intuition
Matrix multiplication depends on row–column pairing.
Transposition swaps this pairing, which reverses the order.
5. Transpose of the Identity Matrix #
\[ I^T = I \]Intuition
The identity matrix is symmetric about its main diagonal.
6. Transpose of a Zero Matrix #
\[ 0^T = 0 \]Intuition
All entries are zero, so swapping rows and columns changes nothing.
7. Transpose and Inverse (Invertible Matrices) #
\[ (A^{-1})^T = (A^T)^{-1} \]Intuition
Inversion and transposition commute for invertible matrices.