Determinants

To find the determinant of the matrix , we will use the concept of cofactor expansion along the first row:

Calculate each of these 2x2 determinants:

Substitute these results back into the original expression:

Expanding and simplifying each part yields:

Notice the trigonometric identities . Substituting these into the expression gives:

The determinant simplifies to , which is independent of and depends solely on . Hence, the determinant is independent of only.

To solve the problem, we need to evaluate the determinant of the given matrix:

We are asked to find the value of given that this determinant is zero. The matrix represents a system where the columns are polynomials in , , and respectively. Each row can be treated as a polynomial function of a variable. If the determinant of a matrix composed of polynomials is zero, it implies that the rows (or columns) of the matrix are linearly dependent, meaning one row can be expressed as a combination of others.

For the polynomial vector formed by rows being dependent, a Vandermonde matrix comes in play:

The simplified difference for a determinant with added terms like compared to a normal polynomial matrix is zero when expanded due to symmetric polynomial properties.

Given the condition that the determinant is zero and , , are distinct, the expression:

Since , , and are distinct, . Therefore, for the determinant to be zero, at least one of put (as expressions with xyz factors) must equate to zero which implies due to symmetry complexes in its multi-linear form when crossed with polynomial roots.

Therefore, the value of is:

To find the value of the determinant , let's perform a series of elementary row operations that simplify the process. The determinant can be calculated using any of its rows or columns. Let's simplify the calculations by using row operations:

We know that a determinant's value remains unchanged if we add a scalar multiple of one row to another row.

Step 1: Subtract Row 3 from Row 1.

Step 2: Subtract Row 3 from Row 2.

Now, observe that after these transformations, the first two rows are linearly dependent:

Row 1 + Row 2 =

This means the determinant is zero because the presence of linearly dependent rows (or columns) causes the determinant of a matrix to be zero.

Thus, the value of the determinant is 0.

We are given that is a matrix and . We need to find .
From matrix theory, we know the following property of determinants for an inverse matrix: Substitute the given value :
Thus, the determinant of is .

Given and , both matrices are of order 3. Using properties of determinants: where is the scalar matrix times the identity matrix of order 3. Therefore,

We are given that is a matrix and . We need to find . From matrix theory, we know the following property of determinants for an inverse matrix: Substitute the given value : Thus, the determinant of is .