Matrices And Determinants

Step 1: Properties
A and B are symmetric, so and .
Step 2: Transpose Calculation
Consider . Using , we get .
Step 3: Substitution
Substitute and : .
Step 4: Conclusion
. Since the transpose is the negative of the original matrix, it is skew-symmetric.
Final Answer: (b)

Step 1: Concept
A matrix has rank 3 if and only if its determinant is non-zero.
Step 2: Analysis
Calculate . Applying and factoring out , we get .
Step 3: Evaluation
Simplifying the determinant using row operations and , we get .
Step 4: Conclusion
Thus, and .
Final Answer: (d)

Step 1: Concept
By definition, for a skew-symmetric matrix , .
Step 2: Analysis
For the diagonal elements where , we have , which implies , so .
Step 3: Evaluation
The diagonal elements of any skew-symmetric matrix are always zero.
Step 4: Conclusion
The trace of a matrix is the sum of its diagonal elements. Therefore, Trace of .
Final Answer: (c)

Step 1: Concept
An orthogonal matrix satisfies .
Step 2: Analysis
Taking the determinant on both sides: . Since and , we have .
Step 3: Evaluation
Using the property , the equation becomes .
Step 4: Conclusion
Solving for , we get .
Final Answer: (c)

Step 1: Understanding the Concept:
is orthogonal rotation matrix. .

Step 2: Detailed Explanation:
. Then .
.
is a shear matrix. .
So .

Step 3: Final Answer:
Option (B).

Step 1: Understanding the Concept:
If is a square root of , then .

Step 2: Detailed Explanation:
Let . Then . For , we need and off-diagonals 0 which are already 0. So , or .

Step 3: Final Answer:
, which corresponds to option (B).

Step 1: Understanding the Concept:
Expand the determinant and substitute .
Step 2: Detailed Explanation:


This simplifies to
Step 3: Final Answer:
None of the given options match.

Concept: Matrix multiplication: Row × Column.

Step 1: Multiply first row.

Step 2: Multiply second row.