Demonstrating that if A has linearly independent columns, (transpose of A) is invertible. Therefore, the transpose of a invertible matrix is itself invertible.

The transpose of a matrix is an operator in linear algebra that flips a matrix over its diagonal; that is, it reverses the row and column indices of matrix A by creating **another matrix**, generally represented by AT (among **other notations**). The transpose of a square matrix is unique.

A transposed* matrix is one that has been flipped over its diagonal. That is, if the element in position (row, column) of A is called "a", then the element in position (column, row) of **its transpose** is called "a". For example, taking the matrix:

$$\begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}$$

Its transpose is:

$$\begin{bmatrix} 4 & 3 \\ 2 & 1 \end{bmatrix}$$

Since these elements are just numbers, their product is also a number: it's 42.

Transposing a non-square matrix will result in a matrix with zero on its diagonal. Transposing a square matrix more than once will always return the same matrix.

TRANSPOSE renders a vertical range of cells as a horizontal range or vice versa. The TRANSPOSE function must be entered as an array formula in a range that has **the same number** of rows and columns as the source range has.

Function TRANSPOSE

- Select blank cells. First select some blank cells.
- Type =TRANSPOSE With those blank cells still selected, type: =TRANSPOSE
- Type the range of the original cells. Now type the range of the cells you want to transpose.
- Finally, press CTRL+SHIFT+ENTER.

In addition, = Inversion and inverse systems: If a system produces unique output signals from separate input signals, it is said to be invertible. If an invertible system generates the output for the input, then its inverse does as well: it is also invertible.

Now consider the following example. Suppose that we have an invertible system with two inputs and one output. Then there must be at least two distinct ways of writing down the output for each pair of inputs. Let's call these pairs of inputs "a" and "b". We can write down how the output will be for any given value of "a" using a table. There are as many rows in the table as there are values of "a". Each row corresponds to a different value of "b".

Similarly, we can write down how the output will be for any given value of "b" using another table. Again, there are as many columns in the table as there are values of "b". Each column corresponds to a different value of "a". Now suppose that we want to find out what output will be for some particular values of "a" and "b". We can look up the corresponding values in the tables above and read off the desired information. This process is called "computing the output for given inputs".