1. What is the Matrix Transpose?
The transpose of a matrix \(A\), denoted \(A^T\), is formed by flipping \(A\) over its main diagonal. Rows become columns and columns become rows.
$$A=[a_{ij}]\quad\Rightarrow\quad A^T=[a_{ji}]$$
If \(A\) is an \(m\times n\) matrix, then \(A^T\) is an \(n\times m\) matrix.
2. Formula & Quick Rules
- Definition: \( (A^T)_{ij} = A_{ji} \).
- Dimension rule: if \(A\) is \(m\times n\), then \(A^T\) is \(n\times m\).
- Transpose of a product: \( (AB)^T = B^T A^T \).
- Transpose of a transpose: \( (A^T)^T = A \).
- Transpose of a scalar multiple: \( (cA)^T = c A^T \).
- Symmetric matrices: \( A = A^T \) (only in this case the transpose equals the original matrix).
3. Worked Examples
Example 1 — 2×2 matrix
$$A=\begin{bmatrix}1 & 4 \\
2 & 5\end{bmatrix}$$
Transpose:
$$A^T=\begin{bmatrix}1 & 2 \\
4 & 5\end{bmatrix}$$
Example 2 — 3×2 matrix
$$B=\begin{bmatrix}3 & -1 \\
0 & 2 \\
5 & 4\end{bmatrix}$$
Transpose:
$$B^T=\begin{bmatrix}3 & 0 & 5 \\
-1 & 2 & 4\end{bmatrix}$$
4. Common Mistakes
- Confusing transpose with inverse: Transpose flips indices; inverse undoes multiplication.
- Not switching dimensions: Remember \(m\times n\rightarrow n\times m\).
- Forgetting signs: Negative entries keep their sign when transposed.
- Assuming all matrices are symmetric: Only symmetric matrices satisfy \(A=A^T\).
5. Practice Problems
Try the problems below. Click "Show Answer" to reveal the transpose.
Exercise 1
$$\begin{bmatrix}1 & 3 \\
4 & 2\end{bmatrix}$$
Show Answer
$$\begin{bmatrix}1 & 4 \\
3 & 2\end{bmatrix}$$
Exercise 2
$$\begin{bmatrix}0 & -2 & 5\end{bmatrix}$$
Show Answer
$$\begin{bmatrix}0 \\
-2 \\
5\end{bmatrix}$$
Exercise 3
$$\begin{bmatrix}7 \\
-1 \\
4\end{bmatrix}$$
Show Answer
$$\begin{bmatrix}7 & -1 & 4\end{bmatrix}$$
Exercise 4
$$\begin{bmatrix}2 & 0 \\
-3 & 5 \\
4 & 1\end{bmatrix}$$
Show Answer
$$\begin{bmatrix}2 & -3 & 4 \\
0 & 5 & 1\end{bmatrix}$$
Transpose operations often appear together with eigenvalues, matrix rank, and QR decomposition.
