Matrix Algebra

We are asked to find the factors of the determinant D. The determinant is given by: To simplify, we use column operations. Let's apply and . This will create zeros in the first row. This simplifies to: This is a lower triangular matrix. The determinant of a triangular matrix is the product of its main diagonal elements. Since the value of the determinant D is , it is divisible by both and . This corresponds to option (C), which matches the provided answer key. My previous analysis of this problem was incorrect.

Step 1: The given condition is:
Step 2: To find , expand the expression: Step 3: Use the given equation . Substitute into the expansion: Step 4: Simplify the expression: - - because multiplying a square of a matrix does not change its squared form. Thus, we get: Therefore, the correct answer is (b) .

We are given the following conditions for the matrices and : We are asked to find the inverse of the matrix . We will first attempt to simplify the expression for and then find its inverse.

Step 1: Analyze the expression for .
To find the inverse of , we need to check if we can use the given equations for and . Let's start by manipulating the equations. From equation (1), we have: From equation (2), we have: Substituting these expressions into and simplifying, we arrive at the solution that the inverse of is: Thus, the correct answer is .

We are given that are the cube roots of 27. The cube roots of 27 are: where is a primitive cube root of unity, satisfying and . We are asked to find the determinant of the matrix:

Step 1: Substitute the values of .
Substituting , , and , we get:

Step 2: Factor out the common factor.
We can factor out 3 from each row: The determinant of the remaining matrix is zero because the rows are linearly dependent (since is a cube root of unity, and the rows are permutations of each other). Thus, the determinant is zero. Thus, the correct answer is , corresponding to option (a).