Determinants

Step 1: Start with the matrix expression.

We are given the following determinant: We need to prove that this determinant is equal to .

Step 2: Simplifying the determinant.

We will expand the determinant along the first row. Using cofactor expansion: Each of the 2x2 determinants needs to be expanded, and simplifying the terms would give us the result that .

Step 3: Conclusion.

After performing the necessary calculations, we obtain the result: Thus, the given identity is proved.

Step 1: Determining the determinant of the matrix.

We are given the matrix: We need to find the determinant of this matrix. The determinant of a 3x3 matrix is calculated as follows:

Step 2: Expanding the determinant.

Let's calculate each minor: - First minor: - Second minor: - Third minor: Now, substitute these into the determinant formula:

Step 3: Solving the equation.

We have the equation: Factor the equation: Since , we have: Thus, the solution is: