Solution1: Given Let’s simplify.

Solution1: Given
Let’s simplify
Use DeMoivre’s Theorem we get
Hence,
Solution 2:
Simplify further we get
Solution 3: Given
i):
So,
ii):

iii):
We know that if and then
So,
Hence,
iv):
And
So,
Solution 4: Given matrices
So,
And
Solution 5: Given matrix is
The augmented matrix with identity is
Let’s perform elementary operations

Solution 6: The system of equations is
The augmented matrix is
Let’s perform elementary row operations.
From the last matrix, the general solution is
Where is any real number, this means that the given system has
infinitely many solutions.
Solution 7: Given that matrix A can be expressed as
Where matrices B and C are symmetric and anti – symmetric respectively. That is:
Now,

This implies that
Add equations (1) and equations (2) we get
Subtract equations (1) and equations (2) we get
Solution 8: Given the rotation matrix . To prove
Similarly
Continuing in this way, we get
This completes the proof.