Applications Of Determinants And Matrices

We are given a matrix , where each row sum is equal to 6 and .

Step 1: Construct the Matrix Using Given Conditions
From the row sum condition: Using the given condition that for , we have: For : For : Since , we substitute: For , we use:

Step 2: Solve for Matrix Elements
From row sums: Using , we get: Similarly, using : Solving, we find: For : Using , we substitute:

Step 3: Compute Determinant
The determinant of a matrix is given by: Thus, the determinant is .

We are given the determinants and , as well as the ratio . Our goal is to compute .

Step 1: We begin by simplifying the determinant . First, evaluate :

We expand this determinant along the first row:

Now calculate the 2×2 determinants:

Substituting these values back into the expression for :

Step 2: Next, simplify the determinant :

Expanding along the first row:

Now calculate the 2×2 determinants:

Substituting these into :

Step 3: Given the ratio , we can now solve for .

By properties of determinants, the solution simplifies to:

Step 1: Understanding the Given Determinant

We are given the determinant:

This determinant is a circulant determinant, which has a standard property:

Step 2: Applying the Property to Our Determinant

Substituting , , :

Since the given equation has roots related by cube roots of unity, we use properties of symmetric sums.

Step 3: Evaluating the Determinant

From determinant properties of circulant matrices, we conclude:

We are given the properties of the adjugate matrices of and , and we need to determine .

Step 1: Property of Determinant of Adjugate
For any square matrix , the determinant of the adjugate is given by: where is the order of the matrix.

Step 2: Using the Determinant Multiplication Rule
Since is the product of two matrices, we use the property: Applying the determinant property: Thus,

Step 3: Expressing in Terms of Given Values
We know that: Multiplying these equations: So,

Step 4: Finding the Inverse

We need to evaluate the determinant:

Step 1: Expand the Determinant Along the First Row

Using the determinant expansion along the first row:

Step 2: Compute the 2×2 Determinants

Each determinant is evaluated as follows:

Step 3: Compute the Final Value

After simplifications, we obtain: