Matrix transpose

Let's take an matrix A. If the matrix's transpose is B, then the dimensions of B are , such that: . Here is the matrix A:

Then, the matrix B is as given:

Essentially, we can think of this as writing the columns of A as the rows of the transposed matrix, B.

We usually write the transpose of A as A^T.

A symmetric matrix is a special kind of matrix. It is an n×n matrix that, when transposed, is exactly the same as before we transposed it.

The following are the properties of inverses and transposes:

If A is an invertible matrix, then so is A^T, and so (A^-1)^T = (A^T)^-1 = A^-T.