Determinants

Let


clearly rank of A is 1 if a = -6

Step 1: Understanding the problem.
We are given a determinant of a 3x3 matrix with the elements: The goal is to find the value of . First, notice that the determinant is set equal to zero, meaning that the rows of the matrix must be linearly dependent.
Step 2: Expanding the determinant.
We expand the determinant of a 3x3 matrix as follows: Now we compute the 2x2 determinants. First, for the first 2x2 determinant: The second 2x2 determinant: Thus, the determinant simplifies to zero as expected.
Step 3: Conclusion.
Since the determinant is zero, the rows are linearly dependent. Through further algebraic manipulation, we find that the value of . Thus, the correct answer is (b) -1.

The correct answer is A:17
Given;
If ,then
New trace of A=sum of diagonal elements of A

Given,
[Applying


which is independent of

The given matrix is
using

points are collinear i.e., area of triangle =0 )

So, the rank of matrix is always less than .

On solving the determinant,

Step 1: Simplifying the Determinant.
The determinant is a 3x3 matrix. To solve for , we need to evaluate the sum of the determinants for each value of . The entries depend on and , and we need to simplify the expression.
Step 2: Using Properties of Determinants.
We can use properties of determinants (such as expansion along rows or columns) to simplify the matrix. However, after simplification and summing over all values of , the result for is found to be zero.
Conclusion.
The correct answer is (2) , as the sum of the determinants evaluates to zero.

Step 1: Understanding the determinant.
The given matrix is a 3x3 determinant. We need to evaluate the determinant first and then integrate it.
Step 2: Expanding the determinant.
We can expand the determinant using cofactor expansion or a suitable method to simplify it. After simplification, we find that simplifies to a function that can be integrated.
Step 3: Performing the integration.
After evaluating the determinant, we integrate the resulting function from to .
Conclusion.
The value of the integral is , which corresponds to option (4).

The system has non- zero solution, if

Step 1: Use cofactor expansion or any method to compute the determinant. The determinant is given by: Step 2: After calculation, we find .

Final Answer: