Let `Ma n dN` be two `3xx3` non singular skew-symmetric matrices such that `M N=N Mdot` If `P^T` denote the transpose of `P ,` then `M^2N^2(M^T N^(-1))^T` is equal to `M^2` b. `-N^2` c. `-M^2` d. `M N`

Options

A. `M^(2)`
B. `-N^(2)`
C. `-M^(2)`
D. `MN`

Explanation

To solve the problem step by step, we need to analyze the expression \( M^2 N^2 (M^T N^{-1})^T \) given the properties of the skew-symmetric matrices \( M \) and \( N \).

### Step 1: Understand the properties of skew-symmetric matrices
Since \( M \) and \( N \) are skew-symmetric matrices, we have:
\[ M^T = -M \quad \text{and} \quad N^T = -N \]

### Step 2: Simplify the expression
We need to simplify the expression \( M^2 N^2 (M^T N^{-1})^T \). Let's first rewrite \( (M^T N^{-1})^T \):
\[ (M^T N^{-1})^T = (N^{-1})^T (M^T)^T = N^{-T} M^{-T} \]
Using the properties of skew-symmetric matrices:
\[ N^{-T} = -N^{-1} \quad \text{and} \quad M^{-T} = -M^{-1} \]
Thus, we can rewrite:
\[ (M^T N^{-1})^T = (-N^{-1})(-M^{-1}) = N^{-1} M^{-1} \]

### Step 3: Substitute back into the expression
Now substituting this back into our expression:
\[ M^2 N^2 (M^T N^{-1})^T = M^2 N^2 (N^{-1} M^{-1}) \]

### Step 4: Rearranging the terms
We can rearrange the expression:
\[ = M^2 N^2 N^{-1} M^{-1} \]
Since \( N^2 N^{-1} = N \):
\[ = M^2 N M^{-1} \]

### Step 5: Use the commutation property
Given that \( MN = NM \), we can replace \( NM \) with \( MN \):
\[ = M^2 MN M^{-1} \]
Now, using the property of inverses:
\[ = M^2 M N = M^3 N \]

### Step 6: Final simplification
Now, we know that \( M^3 = M \cdot M^2 \) and since \( M \) is skew-symmetric, we can express \( M^3 \) in terms of \( -M^2 \):
\[ = -M^2 \]

### Conclusion
Thus, the final result is:
\[ M^2 N^2 (M^T N^{-1})^T = -M^2 \]

### Answer
The answer is option **c. \(-M^2\)**.