Matrix

To determine the condition for given the inequality:

we first compute the determinant of the matrix:

Expanding the determinant along the first row:

Calculating the determinants:

Substituting these back into the expression:

Simplifying:

Thus, the inequality becomes:

Rearranging terms:

To find a lower bound for , we consider the case when . Let . Substituting into the inequality:

Rearranging:

Factoring the cubic equation:

The roots of the equation are (double root) and . The inequality holds for and .

For , is satisfied. For , the inequality is not satisfied. Therefore, the minimum value of occurs when :

Thus, the inequality must hold.

Therefore, the correct option is:

We are given that the matrix is a singular matrix. For a matrix to be singular, its determinant must be zero.
Step 1: The determinant of matrix is: Since the matrix is singular, the determinant is zero: Solving for :
Step 2: Now, we need to find the quadratic equation whose roots are and . For a quadratic equation, if the roots are and , the equation can be written as: Substitute and : Thus, the quadratic equation is: Multiplying through by 6 to eliminate the fraction:
Conclusion: Therefore, the quadratic equation having the roots and is .

Step 1: Apply Row or Column Operations
We'll apply column operations to simplify the determinant.

The determinant becomes:

Expanding along the first row:

Step 2: Factorize Terms Using Algebraic Identities
Using the factorization identities:




Thus,

Step 3: Factor Out Common Terms
Factoring out common terms:

Step 4: Evaluate the Remaining Determinant
Expanding the remaining determinant:

Expanding each term:

Expanding fully:

Upon simplifying, this reduces to:

Step 5: Final Answer

Final Answer: (4)

Step 1: Expand the matrix equation:

This gives us:

Now, equate the scalar part and the matrix part separately.

Step 2: The equation on the right-hand side is just a scalar, which implies that the matrix part must be zero, i.e., the matrix must contribute nothing to the sum. Thus, we can ignore the matrix for now.

Solving for , we get:

Step 3: Now, substitute the value of into the matrix equation. The matrix part must satisfy:

This gives us:

Solving this equation, we find that .

Step 4: Now that we have and , we can compute :

Hence, the value of is approximately 10, and thus the correct answer is .

We are given that is a matrix and , and we are to determine if .
- From the reason (R), we know that for any square matrix of order , the determinant of its adjugate matrix is given by: - For being a matrix ( ), we apply the formula: So, the assertion (A) that is false. Thus, (A) is false but (R) is true.

We are given the matrix .

To find , we first compute the transpose of the matrix :

Now, subtract from :

The determinant of this matrix is , since it is a skew-symmetric matrix (i.e., its transpose is equal to its negative), and the determinant of a skew-symmetric matrix of odd order is always .

Thus, .

To determine the rank of the matrix, we analyze the system of equations: The system has only the trivial solution ( ) if and only if the determinant of the coefficient matrix is non
-zero. This implies that the matrix is full rank, meaning its rank is equal to the number of rows (or columns), which is 3.
The matrix is:

If the determinant of this matrix is non
-zero, the rank is 3. If the determinant were zero, the rank would be less than 3, and the system would have infinitely many solutions (non
-trivial solutions). Since the problem states that the system has only the trivial solution, the rank must be 3. Thus, the correct answer is:

Step 1: Subtract the first row from the second row:

Step 2: Subtract twice the first row from the third row:

Step 3: Subtract the second row from the third row:

For the system to have no solution, the third row must be inconsistent, meaning we need the last entry in the third row to be nonzero while the coefficient of is zero. Thus, for inconsistency:

Thus, the value of that makes the system inconsistent is .