Transpose of a matrix  Matrix transformations  Linear Algebra  Khan Academy  Summary and Q&A
TL;DR
Matrix transpose involves swapping the rows and columns of a matrix, resulting in a new matrix with the dimensions flipped.
Questions & Answers
Q: What is matrix transpose?
Matrix transpose is the operation of swapping the rows and columns of a matrix, resulting in a new matrix with the dimensions flipped.
Q: How is matrix transpose denoted?
The transpose of a matrix A is typically denoted as A^T or A with superscript T.
Q: Do all matrices have a transpose?
Yes, all matrices can have a transpose, although the dimensions of the resulting transposed matrix will be different from the original.
Q: What is the relationship between matrix transpose and matrix dimensions?
The transpose of an m x n matrix will result in an n x m matrix. The number of rows in the original matrix becomes the number of columns in the transposed matrix, and vice versa.
Summary & Key Takeaways

Matrix transpose involves swapping the rows and columns of a matrix.

The transpose of an m x n matrix results in an n x m matrix.

The entries of the original matrix become columns in the transposed matrix.