** Transpose vs Inverse Matrix
**

The transpose and the inverse are two types of matrices with special properties we encounter in matrix algebra. They are different from each other, and do not share a close relationship as the operations performed to obtain them are different.

They have wide applications in the field of linear algebra and the derived implementations such as computer science.

**More about Transpose Matrix**

Transpose of a matrix *A* can be identified as the matrix obtained by rearranging columns as rows or rows as columns. As a result, each element’s indices are interchanged. More formally, transpose of matrix *A*, is defined as

where

In a transpose matrix, the diagonal remains unchanged, but all the other elements are rotated around the diagonal. Also, the size of the matrices also changes from m×n to n×m.

The transpose has some important properties, and they allow easier manipulation of matrices. Also, some important transpose matrices are defined based on their characteristics. If the matrix is equal to its transpose, then the matrix is symmetric. If the matrix is equal to its negative of the transpose, the matrix is a skew symmetric. The conjugate transpose of a matrix is the transpose of the matrix with the elements replaced with its complex conjugate.

**More about Inverse Matrix**

Inverse of a matrix is defined as a matrix which gives the identity matrix when multiplied together. Therefore, by definition, if *AB = BA = I* then *B* is the inverse matrix of *A* and *A* is the inverse matrix of *B*. So, if we consider *B* = *A*^{-1} , then *AA*^{-1 }= *A*^{-1}*A = I*

For a matrix to be invertible, the necessary and sufficient condition is that the determinant of *A* is not zero; i.e |*A*| = det(*A*) ≠ 0. A matrix is said to be invertible, non-singular, or non–degenerative if it satisfies this condition. It follows that *A* is a square matrix and both *A*^{-1} and *A* has the same size.

The inverse of the matrix *A* can be calculated by many methods in linear algebra such as Gaussian elimination, Eigendecomposition, Cholesky decomposition, and Carmer’s rule. A matrix can also be inverted by block inversion method and Neuman series.

**What is the difference between Transpose and Inverse Matrix?**

• Transpose is obtained by rearranging the columns and rows in the matrix while the inverse is obtained by a relatively difficult numerical computation. (But in reality both are linear transformations )

• As a direct result, the elements in the transpose only change their position, but the values are the same. But in the inverse, the numbers can be completely different from the original matrix.

• Every matrix can have a transpose, but the inverse is defined only for square matrices, and the determinant has to be a non-zero determinant.

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