Linear Algebra

To determine which statement is NOT correct, let's analyze the given system of linear equations:

We need to check the conditions under which the system has a unique solution, infinite solutions, or is inconsistent. We will consider each case as per the given options:

1. The system has a unique solution if and :

For the system to have a unique solution, the determinant of the coefficient matrix must be non-zero. The coefficient matrix is:

Calculating the determinant, we have:

Continuing,

The determinant simplifies to

The determinant is zero when , leading to no unique solutions. Hence, if , the determinant is non-zero, and the system may have a unique solution, given that does not affect determinant independence. Thus, this statement may be correct.

1. The system has an infinite number of solutions if and :

If , the determinant is zero ( ), indicating possible infinite solutions. To check if these exist, substitute and in the equations and verify consistency:

• Substitute in the equations:

Checking compatibility, for , the intervention proves consistent.

1. The system is inconsistent if and :

For , the determinant is zero. To check inconsistency, see if multiplying equations doesn't provide a valid propagation:

• Given equation manipulation for shows disagreement; hence inconsistency holds true.

1. The system is consistent if :

If , the determinant is non-zero, and hence, the system is logically consistent.

After evaluating all statements, it's verified that option 3 (The system is inconsistent if and ) is NOT correct.

We are given the system of equations:
- We can solve this system by using elimination or substitution to obtain the conditions under which the system has no solution. For a system to have no solution, the determinant of the coefficient matrix must be zero, or the equations must be inconsistent.
- After solving the system, we find that the system will have no solution when and .

Conclusion: The correct answer is (1), as the system has no solution when and .

We are given a system of three linear equations and told that it has infinitely many solutions. We need to find the value of the expression .

The system of equations is:

Concept Used:

For a system of linear equations , the condition for having infinitely many solutions is that the determinant of the coefficient matrix , denoted as , must be zero, and the determinants of the matrices formed by replacing each column with the constant vector , denoted as , must also be zero.

Step-by-Step Solution:

Step 1: Set the determinant of the coefficient matrix ( ) to zero.

The coefficient matrix is:

The determinant is:

Expanding along the first row:

Step 2: Set the determinant to zero.

To find , we replace the second column of with the constant vector :

Expanding along the first row:

Step 3: Substitute the value of into equation (1) to find .

Using in the equation :

To eliminate the fraction, multiply the entire equation by 11:

Final Computation & Result:

We have found the values and . Now we compute the value of :

Thus, the value of is 31.

No. of elements in :

No. of elements in : since no zero in

No. of elements in :

or

or

So, it simplifies to

Thus,

Step 1: Analyze each set.

includes symmetric matrices, so elements above the diagonal determine the matrix. With 5 choices for each, and 6 such positions: includes skew-symmetric matrices, where non-diagonal elements are independent, and diagonal elements must be 0 (which are not in ), invalidating . Thus: must balance the trace to be zero. Choosing two elements freely allows the third to be determined:

Step 2: Calculate the union of sets.

Using the inclusion-exclusion principle, find :

To determine the value of for the given system of equations having infinitely many solutions, we need to ensure the system is consistent and dependent. Such a system results in at least one redundant equation. Below is how we approach the problem:

1. Consider the system of equations:
2. For the system to have infinitely many solutions, the determinant of the coefficients must be zero. Construct the coefficient matrix:

1. Calculate the determinant of this matrix and set it equal to zero:

1. Calculate each of the 2x2 determinants:
   • First 2x2 determinant:

• Second 2x2 determinant:

• Third 2x2 determinant:

1. Substitute these into the determinant equation:

1. Simplify the equation:

1. Re-arrange the terms:

1. From this, express it in terms of :

1. On comparison, the coefficients balance out when and . Hence, .

The correct answer is therefore 57.

includes symmetric matrices, so elements above the diagonal determine the matrix. With 5 choices for each, and 6 such positions:

includes skew-symmetric matrices, non-diagonal elements are independent, and diagonal elements must be 0 (not in ), invalidating .

must balance the trace to be zero. Choosing two elements freely allows the third to be determined:

Using the inclusion-exclusion principle, find :

Step 1: First, calculate the powers of the matrix . Use matrix multiplication to compute , , and higher powers as necessary.
Step 2: Solve for the set by substituting the appropriate powers of into the given condition , and identify the values of .
Step 3: Finally, compute , which is the number of elements in the set . Thus, the final value of is found.

For the hyperbola , the length of the latus rectum is given by . The length of the latus rectum for is , so we have: Similarly, for , the length of the latus rectum is , and the given length is , so: Now, we are given that the product of the lengths of their transverse axes is , so: From these equations, we solve for and then compute .
Final Answer: .