Matrices And Determinants

Step 1: Analyze the matrix multiplication.
We use matrix multiplication to find the result of .
Step 2: Conclusion.
Thus, the correct result is .
Final Answer:

The trace of a square matrix is the sum of the elements on the main diagonal (the diagonal from the top left to the bottom right). For the given matrix , the main diagonal elements are: So, the trace of matrix is:
Step 2: Conclusion.
Therefore, the trace of matrix is , and the correct answer is (a).

Step 1: Evaluating the determinant.
To evaluate this determinant, apply cofactor expansion. The value of the determinant simplifies to .

Step 2: Conclusion.
The correct answer is , corresponding to option (2).

Step 1: Understanding collinearity.
When three points are collinear, the rank of the matrix formed by these points will always be 2, as they lie on a straight line.

Step 2: Conclusion.
The correct rank is 2, corresponding to option (2).

Step 1: Analyzing the determinant.
Since the rows and columns of the determinant are linear, we know that the determinant of the matrix will be zero when , , and are in arithmetic progression.

Step 2: Conclusion.
The determinant equals zero, corresponding to option (1).

Step 1: Understanding the formula.
For any square matrix , the relation holds, where is the identity matrix and is the determinant of matrix . Step 2: Analyzing the options.
- (1) : This is incorrect because does not give .
- (2) : This is correct as per the formula .
- (3) : This is a repetition of option (2) and incorrect in this context.
- (4) None of these: This is incorrect as option (2) is correct.
Conclusion.
The correct answer is (2) , as the determinant of matrix is the result when multiplying by its adjugate matrix.

To find the adjoint of matrix , we calculate the cofactor matrix and then transpose it.

To find the rank of matrix , we calculate its determinant. Since the determinant is non-zero, the matrix has rank 1.

Step 1: Matrix Multiplication.
Using the distributive property of matrices and the fact that , we find: Thus, the expression simplifies to .
Step 2: Conclusion.
The correct answer is (B), .

To find the determinant, use the matrix determinant formula.
Final Answer:

By calculating the determinant or using row operations, we find that the rank of the matrix is 1 when .
Final Answer:

Step 1: Set the values in the equation .
Step 2: Solve for and find that .

Final Answer:

Step 1: Check if , where is the transpose of and is the identity matrix.
Step 2: Calculate the transpose and product of , which is found to be the identity matrix.

Final Answer:

When multiplying matrices and , the result is the zero matrix , indicating that the matrices are orthogonal.
Final Answer:

By solving the system of equations, we find that there is a unique solution for the values of , , and .
Final Answer: