Transpose of a Matrix

In matrix algebra, the transpose of a matrix is one of the most basic and frequently used operations.

It is widely used in coordinate geometry, systems of equations, transformations, and data representation in higher mathematics and science.

The transpose of a matrix is obtained by interchanging its rows and columns.

• If a matrix is A, its transpose is denoted by A^T

• If A is of order m × n, then
  A^T is of order n × m

• First row of A becomes first column of A^T, and so on.

1. Transpose of Transpose
   \[(A')' = A \quad \text{or} \quad (A^T)^T = A\]
   Taking transpose twice gives back the original matrix.

2.  Scalar Multiplication
   \[(kA)' = kA' \quad \text{or} \quad (kA)^T = kA^T\]
   Scalar can be taken out unchanged.

3. Transpose of Sum
   \[(A + B)' = A' + B' \quad \text{or} \quad (A + B)^T = A^T + B^T\]
   Transpose distributes over addition.

4. Transpose of Product (Very Important)
   \[(AB)' = B'A' \quad \text{or} \quad (AB)^T = B^T A^T\]
   Order reverses in product.

If \[\mathrm{A=}{ \begin{bmatrix} {-2} \\ {4} \\ {5} \end{bmatrix}},\mathrm{B=}{ \begin{bmatrix} {1} & {3} & {-6} \end{bmatrix}},\] verify that (AB)′ = B′A′.

Solution: We have

\[\mathrm{A}= \begin{bmatrix} -2 \\ 4 \\ 5 \end{bmatrix},\mathrm{B}= \begin{bmatrix} 1 & 3 & -6 \end{bmatrix}\]

then \[\mathrm{AB}= \begin{bmatrix} -2 \\ 4 \\ 5 \end{bmatrix} \begin{bmatrix} 1 & 3 & -6 \end{bmatrix}= \begin{bmatrix} -2 & -6 & 12 \\ 4 & 12 & -24 \\ 5 & 15 & -30 \end{bmatrix}\]

Now \[\mathrm{A}^{\prime}= \begin{bmatrix} -2 & 4 & 5 \end{bmatrix},\] \[\mathbf{B}^{\prime}= \begin{bmatrix} 1 \\ 3 \\ -6 \end{bmatrix}\]

\[\mathrm{B^{\prime}A^{\prime}}= \begin{bmatrix} 1 \\ 3 \\ -6 \end{bmatrix} \begin{bmatrix} -2 & 4 & 5 \end{bmatrix}= \begin{bmatrix} -2 & 4 & 5 \\ -6 & 12 & 15 \\ 12 & -24 & -30 \end{bmatrix}=(\mathrm{AB})^{\prime}\]

Clearly (AB)′ = B′A′

• Think of a spreadsheet (like an Excel sheet or Google Sheet) with students as rows and subjects as columns.

• Taking the transpose of that matrix is like rotating the table so that students become columns and subjects become rows.

• Transpose = interchange rows and columns.

• If A is \[m \times n\], then A' is \[n \times m\].

• Standard notation: A' or \[A^T\].

• Key properties: (A')' = A, (kA)' = kA', (A + B)' = A' + B', (AB)' = B'A'.