Determinants

Given ,
and


Now,

Given,
We know that, the sum of the product of element other than the corresponding cofactor
is zero

The derivative is defined as: We are given the following condition: Next, decompose the limit expression: For the limit to exist and be finite, it must be true that as . This implies: Now substitute into the equation: Thus, we find: Final Answer:

To evaluate the determinant of a matrix: Here, Oops! That implies the correct answer should be: Correct Answer (Updated): (B)

We are given the matrix , and we need to find the determinant of matrix . Step 1: Use the formula for the determinant of a 2x2 matrix The determinant of a 2x2 matrix is given by: Step 2: Substitute the values from matrix For the given matrix , we have , , , and . Answer: The determinant of matrix is , so the correct answer is option (2).

Step 1: Recall the formula for the determinant of a 2x2 matrix For a 2x2 matrix , the determinant is given by: Step 2: Apply the formula to the given matrix For the matrix , we have: - , - , - , - . Now, substitute these values into the determinant formula: Answer: Therefore, the value of the determinant is . So, the correct answer is option (4).