Let M and N are two non singular matrices of order 3 with real entries such that `(adjM)=2N` and `(adjN)=M`. If `MN=lambdaI`, then the value the values of `lambda` is equal to (where, (adj X) represents the adjoint matrix of matrix X and I represents an identity matrix)

Answer

To solve the problem, we need to analyze the given equations involving the adjoint of matrices M and N, and their relationship with the identity matrix.

### Step-by-Step Solution:

1. **Understanding the Given Information**:
We have two non-singular matrices \( M \) and \( N \) of order 3 such that:
\[ \text{adj}(M) = 2N \]
\[ \text{adj}(N) = M \]
We also know that:
\[ MN = \lambda I \]
where \( I \) is the identity matrix.

2. **Using the Property of Determinants**:
Recall that for any matrix \( A \) of order \( n \):
\[ \text{det}(\text{adj}(A)) = \text{det}(A)^{n-1} \]
Since \( M \) and \( N \) are 3x3 matrices, we have:
\[ \text{det}(\text{adj}(M)) = \text{det}(M)^2 \]
\[ \text{det}(\text{adj}(N)) = \text{det}(N)^2 \]

3. **Finding Determinants from the First Equation**:
From \( \text{adj}(M) = 2N \), we take the determinant:
\[ \text{det}(\text{adj}(M)) = \text{det}(2N) = 2^3 \cdot \text{det}(N) = 8 \cdot \text{det}(N) \]
Equating the two expressions for \( \text{det}(\text{adj}(M)) \):
\[ \text{det}(M)^2 = 8 \cdot \text{det}(N) \quad \text{(Equation 1)} \]

4. **Finding Determinants from the Second Equation**:
From \( \text{adj}(N) = M \), we take the determinant:
\[ \text{det}(\text{adj}(N)) = \text{det}(M) \]
Thus,
\[ \text{det}(N)^2 = \text{det}(M) \quad \text{(Equation 2)} \]

5. **Substituting Equation 2 into Equation 1**:
Substitute \( \text{det}(M) \) from Equation 2 into Equation 1:
\[ (\text{det}(N)^2)^2 = 8 \cdot \text{det}(N) \]
This simplifies to:
\[ \text{det}(N)^4 = 8 \cdot \text{det}(N) \]
Rearranging gives:
\[ \text{det}(N)^4 - 8 \cdot \text{det}(N) = 0 \]
Factoring out \( \text{det}(N) \):
\[ \text{det}(N)(\text{det}(N)^3 - 8) = 0 \]
Since \( N \) is non-singular, \( \text{det}(N) \neq 0 \), so:
\[ \text{det}(N)^3 = 8 \implies \text{det}(N) = 2 \]

6. **Finding \( \text{det}(M) \)**:
Using Equation 2:
\[ \text{det}(M) = \text{det}(N)^2 = 2^2 = 4 \]

7. **Finding \( MN \)**:
We know:
\[ MN = \lambda I \]
Taking determinants on both sides:
\[ \text{det}(MN) = \text{det}(M) \cdot \text{det}(N) = 4 \cdot 2 = 8 \]
And:
\[ \text{det}(\lambda I) = \lambda^3 \]
Therefore:
\[ \lambda^3 = 8 \implies \lambda = 2 \]

### Conclusion:
The value of \( \lambda \) is \( 2 \).