Transpose vector or matrix

Transpose of \(A\) matrix is \(A^T\) matrix in which switches the row and column indices. If \(A\) is an \(m \times n\) matrix, then \(A^T\) is an \(n \times m\) matrix

\[
{
\begin{bmatrix}
A
\end{bmatrix}
}_{ij}
=
{
\begin{bmatrix}
A
\end{bmatrix}
}^T_{ji}
\]

For example:

\[
A =
\begin{bmatrix}
1 & 2 \\
3 & 4 \\
5 & 6 \\
\end{bmatrix}
\Rightarrow
A^T =
\begin{bmatrix}
1 & 3 & 5 \\
2 & 4 & 6 \\
\end{bmatrix}
\]

Properties

1. \((A^T)^T=A\)

Eg.

\[
A =
\begin{bmatrix}
1 & 2 \\
3 & 4 \\
5 & 6 \\
\end{bmatrix}
\Rightarrow
A^T =
\begin{bmatrix}
1 & 3 & 5 \\
2 & 4 & 6 \\
\end{bmatrix}
\Rightarrow
(A^T)^T =
\begin{bmatrix}
1 & 2 \\
3 & 4 \\
5 & 6 \\
\end{bmatrix} = A
\]

2. \((A + B)^T = A^T + B^T\)

Eg.

\[
A =
\begin{bmatrix}
1 & 2 \\
3 & 4 \\
5 & 6 \\
\end{bmatrix};
B =
\begin{bmatrix}
1 & 2 \\
2 & 1 \\
0 & 2 \\
\end{bmatrix}
\]

\[
\Rightarrow
A + B =
\begin{bmatrix}
2 & 4 \\
5 & 5 \\
5 & 8 \\
\end{bmatrix}
\Rightarrow
(A + B)^T =
\begin{bmatrix}
2 & 5 & 5 \\
4 & 5 & 8 \\
\end{bmatrix}
\]

\[
\Rightarrow
A^T + B^T =
\begin{bmatrix}
1 & 3 & 5 \\
2 & 4 & 6 \\
\end{bmatrix} +
\begin{bmatrix}
1 & 2 & 0 \\
2 & 1 & 2 \\
\end{bmatrix} =
\begin{bmatrix}
2 & 5 & 5 \\
4 & 5 & 8 \\
\end{bmatrix}
\]

3. \((A B)^T = B^TA^T\)

Eg.

\[
A =
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6
\end{bmatrix};
B =
\begin{bmatrix}
0 & 2 \\
4 & 6 \\
3 & 5
\end{bmatrix}
\]

\[
\Rightarrow
AB =
\begin{bmatrix}
1 \times 0 + 2 \times 4 + 3 \times 3 & 1 \times 2 + 2 \times 6 + 3 \times 5 \\
4 \times 0 + 5 \times 4 + 6 \times 3 & 4 \times 2 + 5 \times 6 + 6 \times 5
\end{bmatrix} =
\begin{bmatrix}
17 & 29 \\
38 & 68
\end{bmatrix}
\]

\[
\Rightarrow
(AB)^T =
\begin{bmatrix}
17 & 38 \\
29 & 68
\end{bmatrix}
\]

\[
\Rightarrow
B^TA^T =
\begin{bmatrix}
0 & 4 & 3 \\
2 & 6 & 5
\end{bmatrix} \times
\begin{bmatrix}
1 & 4 \\
2 & 5 \\
3 & 6
\end{bmatrix} =
\begin{bmatrix}
17 & 38 \\
29 & 68
\end{bmatrix}
\]

4. \((cA)^T = cA^T\)

Eg.

\[
c = 2;
A =
\begin{bmatrix}
1 & 4 \\
2 & 5 \\
3 & 6
\end{bmatrix}
\]

\[
\Rightarrow
cA =
\begin{bmatrix}
2 \times 1 & 2 \times 4 \\
2 \times 2 & 2 \times 5 \\
2 \times 3 & 2 \times 6
\end{bmatrix} =
\begin{bmatrix}
2 & 8 \\
4 & 10 \\
6 & 12
\end{bmatrix}
\Rightarrow
(cA)^T =
\begin{bmatrix}
2 & 4 & 6 \\
8 & 10 & 12
\end{bmatrix}
\]

\[
\Rightarrow
A^T =
\begin{bmatrix}
1 & 2 & 3 \\
4 & 5 & 6
\end{bmatrix}
\Rightarrow
cA^T =
\begin{bmatrix}
2 & 4 & 6 \\
8 & 10 & 12
\end{bmatrix}
\]

5. \(\det(A^T)  = \det(A) \)

Read more determinant

Eg.

$$
A =
\begin{bmatrix}
1 & 2 \\
3 & 4
\end{bmatrix}
\Rightarrow
\det(A) = 1 \times 4 – 3 \times 2 = -2
$$
$$
A^T =
\begin{bmatrix}
1 & 3 \\
2 & 4
\end{bmatrix}
\Rightarrow
\det(A^T) = 1 \times 4 – 2 \times 3 = -2 = \det(A)
$$