Matrices

We know that, is always a symmetric matrix.

Given that,
..(i)
and ..(ii)
On multiplying E (i) by 4 and E (ii) by 4 and then subtracting, we get

Now,


Hence, it is a skew symmetric matrix.

Given, .....(i) and ..(ii)
Where and
On solving Eqs. (i) and (ii), we get

and

Given, Now,

Now,

Step 1: Symmetric and Skew-Symmetric Matrix Properties
A matrix is symmetric if .

A matrix is skew-symmetric if .

Step 2: Analyzing the Possibilities
For a matrix to be both symmetric and skew-symmetric, we must have and , which implies .
Therefore, must be the zero matrix, which is a null matrix.
Step 3: Conclusion
Thus, is a null matrix.

Step 1: Understanding Adjoint of a Matrix
The adjugate (or adjoint) of a matrix is the transpose of the cofactor matrix.
For a 3x3 matrix , the inverse is related to the adjugate by the formula:
Step 2: Finding the Determinant of
Since is a rotation matrix, its determinant is 1.
Step 3: Conclusion
Thus, .