Applications Of Determinants And Matrices

The coefficient matrix of the system is:

The determinant is given by:

Simplifying:


Factoring:



Thus, when or .
Case 1:
The system reduces to:

This system has infinitely many solutions.
Case 2:
The determinant of the matrix becomes:

For this case, compute:



Therefore:

Hence:

Case 3:
For this case, the system becomes inconsistent as the determinant , but the equations do not align for the given .
Conclusion: The statement (1) is not correct.

The correct option is (B): Infinitely many solution for

Let = =
for 6 system has unique solution
Now, when



Clearly at = 5, 0 for i = 1, 2, 3
∴ at system has infinite solutions.

We are tasked with analyzing the given expressions and determining the value of . Let us proceed step by step:

1. Determinant of Matrix :
The determinant of matrix is given as:

2. Cofactor Expressions:
The cofactors are computed as follows:

3. Matrix :
The matrix is constructed using the cofactor values:

4. Determinant of Matrix :
The determinant of is calculated as:

5. Scaling :
We are asked to compute :

Final Answer:
The value of is .