A skew symmetric matrix S satisfies the relations `S^(2)+1=0` where I is a unit matrix then SS' is equal to

Options

A. I
B. 2I
C. `-I`
D. none of the above

Answer

To solve the problem, we start with the given condition for the skew-symmetric matrix \( S \):

1. **Given Relation**:
\[ S^2 + I = 0 \]
where \( I \) is the identity matrix.

2. **Rearranging the Equation**:
From the equation above, we can rearrange it to isolate \( S^2 \):
\[ S^2 = -I \]

3. **Understanding Skew-Symmetric Matrices**:
A skew-symmetric matrix \( S \) satisfies the property:
\[ S^T = -S \]
where \( S^T \) is the transpose of \( S \).

4. **Finding \( SS^T \)**:
We need to find \( SS^T \). Since \( S \) is skew-symmetric, we can substitute \( S^T \) with \( -S \):
\[ SS^T = S(-S) = -S^2 \]

5. **Substituting \( S^2 \)**:
Now we substitute \( S^2 \) from our earlier result:
\[ SS^T = -(-I) = I \]

6. **Conclusion**:
Therefore, we conclude that:
\[ SS^T = I \]

### Final Answer:
\[ SS^T = I \]