Question

Is the rank of a matrix equal to the rank of its transpose?

Answer 1

Explaining why the rank of a matrix is equal to the rank of its transpose:

The rank of a matrix is always equal to the rank of its transpose because transpose is just converting a row into a column and vice versa. Therefore, the rank of the matrix does not change.

For example, let's consider a matrix

$$\begin{gathered} A=\left(\begin{array}{cc}2& 6\\ 2& 6\end{array}\right) \\ \left|A\right|=(6\times 2)-(6\times 2) \\ =12-12 \\ =0 \end{gathered}$$

Since $\left|A\right|=0$, the rank is 1.

$$\begin{gathered} A^T=\left(\begin{array}{cc}2& 6\\ 2& 6\end{array}\right) \\ \left|A^T\right|=(6\times 2)-(6\times 2) \\ =12-12 \\ =0 \end{gathered}$$

Since $\left|A^T\right|=0$, the rank is 1

Therefore, the rank of a matrix equal to the rank of its transpose.