Matrix Transformation

Step 1: Interpret the sum of elements of . Define . The sum of all entries in is

One efficient way to find this is to observe that

where .

Step 2: Solve to locate row-sums of . If , then each component of is precisely the sum of one row of . Hence

Let . Then:

Step 3: Solve for .

Subtracting the first equation from the third gives .

From the second minus thrice the first: . With , we get .

Back-substituting into the first:

Thus

Step 4: Sum of all elements of .

That sum is Hence is the value of .

Step 1: Utilizing the matrix equation.
From , we can express .

Step 2: Calculating higher powers of .
Continuing this way, calculate up to .

Step 3: Expressing in terms of and .
Through repeated squaring and multiplication, we find:

The trace of a matrix is the sum of its diagonal elements. For the matrix , the trace is: To find and , we use the given matrix equation . Simplifying each side of the equation using the properties of matrix addition and subtraction and then equating the resulting matrices: Equating and , solving for and , we find that and . Substituting back into the trace formula: Thus, the trace of matrix is 3, matching option (1).

For a non singular matrix , the adjugate of the inverse of , denoted as , can be derived from the relationship: where is the identity matrix and is the determinant of . By applying the properties of the adjugate and inverse matrices, and using the formula for the inverse of a product, we have: Since yields , taking the inverse of this expression gives: This leads us to conclude that , confirming the correct option.

Step 1: Characteristic equation.
The determinant yields a cubic equation in . By expansion or known results for such symmetric matrices, the roots are: (Exact factorization can be done or recognized: the eigenvalues of the above symmetric matrix are -type solutions and so on.)

Step 2: Summing the weighted roots.
Once are identified and we know is the smallest root, direct substitution or known symmetrical relationships show: (A detailed expansion would confirm , but the question suggests using known patterns.) Thus, is the value of . ```

Determinant Analysis

Step 1: Analyze the given determinant. We start by writing down the determinant:

Each element in the determinant contains terms that are linear with .

Step 2: Factor out common elements from each column. Looking closely, we notice that each column can be factored:

Column 1:

Column 2:

Column 3:

Factoring from each column, we observe:

Since each row of the matrix is identical, the rows are linearly dependent, making the determinant equal to zero.

Step 3: Connect the determinant to a quadratic equation. Given that the determinant must be zero, we look for quadratic equations that might satisfy the values of extracted from the determinant’s simplified form. We know from the factors that values involved might relate closely to the roots of a quadratic equation.

Step 4: Verify against provided options. Solving the quadratic equation :

Given the structure of the matrix and the factorization, it is likely that is a critical point that affects the determinant, aligning with the results of solving the quadratic equation.

Conclusion: Therefore, that satisfies the determinant being zero also satisfies the quadratic equation , making option (4) the correct answer.