1. Introduction to Properties of Transpose
The transpose operation follows several neat and predictable rules. These rules make calculations simpler and help in solving matrix equations, simplifying expressions, and understanding symmetric or skew-symmetric matrices.
Below are the most important properties of transpose, each explained clearly with small examples.
2. Property 1: Transpose of a Transpose
Taking the transpose twice brings the matrix back to what it originally was.
\( (A^T)^T = A \)
2.1. Example
A = \begin{bmatrix} 2 & 5 \\ 3 & 7 \end{bmatrix}
Then:
A^T = \begin{bmatrix} 2 & 3 \\ 5 & 7 \end{bmatrix}
Now transpose again:
(A^T)^T = A
3. Property 2: Transpose of a Sum
The transpose of the sum of two matrices is equal to the sum of their transposes.
\( (A + B)^T = A^T + B^T \)
3.1. Example
Let:
A = \begin{bmatrix} 1 & 4 \\ 2 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 5 & 0 \\ -1 & 2 \end{bmatrix}
Then:
A + B = \begin{bmatrix} 6 & 4 \\ 1 & 5 \end{bmatrix}
Transposing both sides confirms the rule.
4. Property 3: Transpose of a Scalar Multiple
Multiplying a matrix by a scalar does not change how transpose works. The scalar comes out unchanged.
\( (kA)^T = kA^T \)
4.1. Example
Let \( k = 3 \) and
A = \begin{bmatrix} 2 & -1 \\ 4 & 5 \end{bmatrix}
Then:
(3A)^T = 3A^T
Both give identical results.
5. Property 4: Transpose of a Product
The transpose of a product reverses the order of matrices.
\( (AB)^T = B^T A^T \)
This is one of the most important properties and appears frequently in proofs and simplifications.
5.1. Example
Let:
A = \begin{bmatrix} 1 & 2 \\ 0 & 3 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 1 \\ -2 & 5 \end{bmatrix}
Compute \( AB \), then take a transpose. Now compute \( B^T A^T \). Both results will match.
6. Property 5: Transpose of a Difference
Like addition, transpose also works neatly with subtraction.
\( (A - B)^T = A^T - B^T \)
6.1. Example
If:
A = \begin{bmatrix} 6 & 3 \\ 1 & 2 \end{bmatrix}, \quad B = \begin{bmatrix} 4 & 1 \\ 0 & 5 \end{bmatrix}
Transposing both sides of \( A - B \) confirms the identity.
7. Property 6: Transpose and Symmetry
Two important matrix types are based on transpose:
- Symmetric matrix: \( A = A^T \)
- Skew-symmetric matrix: \( A^T = -A \)
These depend directly on how transpose behaves.
8. Summary of Properties
All properties at a glance:
- \( (A^T)^T = A \)
- \( (A + B)^T = A^T + B^T \)
- \( (A - B)^T = A^T - B^T \)
- \( (kA)^T = kA^T \)
- \( (AB)^T = B^T A^T \)
These rules make matrix algebra easier and are used constantly in higher topics.