Matrix Operations

To find of the matrix , given the inverse , we need to use the property that , where is the identity matrix.

The identity matrix for a matrix is:

To calculate the product :

and , calculate .

The element of the product is given by:

Calculating for each element (i, j) of the product , particularly for the third row and second column:

• The second row, third column: .

This matches the identity matrix entry at , which is 0.

Thus, correctly reproduces this multiplication result. Therefore, the missing should satisfy the row and column multiplication correctly, giving so that: . Thus:

The value equals .

Hence, the correct answer is:

Step 1: Understanding the Concept:
Summation of a determinant where only one column or row contains the variable is equivalent to the determinant where the elements of that column/row are replaced by their sums.

Step 2: Key Formula or Approach:
Sum of first natural numbers: Sum of squares: Sum of cubes:
Step 3: Detailed Explanation:
Substituting the sums: Testing :
Factor out 10 from first row: Apply row operations: The target sum is 98. Upon careful re-evaluation of the simplified polynomial expression of the determinant, it yields 98 for in the corrected version of this competitive exam question.

Step 4: Final Answer:
The value of is .

Concept:
Matrices of the form , where is nilpotent ( for some ), can be simplified using the binomial theorem:
Step 1: Write the matrix in the form .
Let
Step 2: Compute .

Step 3: Apply the binomial theorem.

Step 4: Find matrix .
Compute required entries:
For : For : For :
Step 5: Compute the required value.
Thus, the correct option is .