Matrices

Step 1: Analyzing the given matrix equation. 
The matrix equation is: 

Step 2: Using the determinant of the matrix. 
The determinant of a matrix is given by: This simplifies to: Step 3: Solving for . 
We are given that the determinant is equal to 0, so: This implies that either or . 
Step 4: Conclusion. 
From , we get . Substituting the values, we find that , corresponding to option (C).

Step 1: Definition of symmetric matrix.
A matrix is symmetric if , i.e., the transpose of is equal to .
Step 2: Definition of skew-symmetric matrix.
A matrix is skew-symmetric if , i.e., the transpose of is equal to the negative of .
Step 3: Combining the properties.
If is both symmetric and skew-symmetric, then: Equating the two, we get: This implies that , or .
Conclusion:
Thus, the statement is True.

Step 1: Understanding the adjugate matrix.
The adjugate (or adjoint) of a matrix , denoted , is the transpose of the cofactor matrix of . The property of the adjugate matrix for an invertible matrix is given by: where is the identity matrix.
Step 2: Applying the property of the adjugate.
For the adjugate of the adjugate, we have the relation: This is a known property of the adjugate matrix, where is the order of the matrix.
Conclusion:
Thus, the statement is True.