transpose

the transpose of a matrix $\text{[math]}$ , denoted by $\text{[math]}$ , is the matrix obtained from $\text{[math]}$ by interchanging rows and columns. specifically, if $\text{[math]}$ is an $\text{[math]}$ -by- $\text{[math]}$ matrix, then $\text{[math]}$ is an $\text{[math]}$ -by- $\text{[math]}$ matrix whose entries are given by the equation
$\text{[math]}$ [cite:;taken from @algebra_axler_2024 definition 3.54]

$\text{[math]}$ this can be generalized to a multiplication of $\text{[math]}$ matrices by induction.

the transpose of an invertible matrix is invertible.