Transpose Of A Matrix

The given determinant is:

The sum of determinants is:

Expanding the determinant for the summation:

Simplify the summations:

, ,

Substitute back:

Perform row operations to simplify:

This gives:

Calculate the determinant:

Simplify:

Solve for :

Factorize:

(as )

Final Answer:

Given the matrix , the determinant is calculated as:

The row transformations are applied as follows:

First, perform the operation and then , resulting in:

The determinant of the modified matrix is:

Next, the adjugate of the matrix is calculated. The formula for the adjugate of a matrix is given by:

For a matrix, , so the formula becomes:

The determinant of the adjugate is calculated as:

Substituting the previously calculated determinant , we get:

Let:

Compute: Thus, is the inverse of .

Now, define: The general expression for is: Using the properties of matrices:

Compute powers of :
Similarly:

Now substitute into : Simplify: Expanding:

Now, compute the sum: Simplifying each term: - - - -

The sum of all elements in the resulting matrix is:

Final Answer: The sum of the elements is 100.