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 find the value of \( \lambda \) given the conditions involving the adjoint matrices of \( M \) and \( N \). Let's break it down step by step.

### Step 1: Understand the properties of adjoint matrices
We know that for any square matrix \( A \) of order \( n \):
\[ \text{adj}(A) = \det(A) \cdot A^{-1} \]
For a 3x3 matrix, we also have:
\[ \det(\text{adj}(A)) = \det(A)^{n-1} = \det(A)^2 \quad \text{(since \( n = 3 \))} \]

### Step 2: Use the given relations
We are given:
1. \( \text{adj}(M) = 2N \)
2. \( \text{adj}(N) = M \)

### Step 3: Take determinants of both sides
Taking the determinant of the first relation:
\[ \det(\text{adj}(M)) = \det(2N) \]
Using the properties of determinants:
\[ \det(\text{adj}(M)) = \det(M)^2 \quad \text{and} \quad \det(2N) = 2^3 \cdot \det(N) = 8 \cdot \det(N) \]
Thus, we have:
\[ \det(M)^2 = 8 \cdot \det(N) \quad \text{(Equation 1)} \]

### Step 4: Take determinants of the second relation
Now, taking the determinant of the second relation:
\[ \det(\text{adj}(N)) = \det(M) \]
Again, using the properties of determinants:
\[ \det(\text{adj}(N)) = \det(N)^2 \]
So we have:
\[ \det(N)^2 = \det(M) \quad \text{(Equation 2)} \]

### Step 5: Substitute Equation 2 into Equation 1
From Equation 2, we can express \( \det(M) \) in terms of \( \det(N) \):
\[ \det(M) = \det(N)^2 \]
Substituting this into Equation 1:
\[ (\det(N)^2)^2 = 8 \cdot \det(N) \]
This simplifies to:
\[ \det(N)^4 = 8 \cdot \det(N) \]
Rearranging gives:
\[ \det(N)^4 - 8 \cdot \det(N) = 0 \]
Factoring out \( \det(N) \):
\[ \det(N)(\det(N)^3 - 8) = 0 \]
Since \( N \) is non-singular, \( \det(N) \neq 0 \), so we can set:
\[ \det(N)^3 - 8 = 0 \implies \det(N)^3 = 8 \implies \det(N) = 2 \]

### Step 6: Find \( \det(M) \)
Using \( \det(N) = 2 \) in Equation 2:
\[ \det(M) = \det(N)^2 = 2^2 = 4 \]

### Step 7: Find \( \lambda \)
We know that:
\[ MN = \lambda I \]
Taking determinants on both sides:
\[ \det(MN) = \det(\lambda I) \]
Using the property of determinants:
\[ \det(M) \cdot \det(N) = \lambda^3 \]
Substituting the values we found:
\[ 4 \cdot 2 = \lambda^3 \implies 8 = \lambda^3 \]
Thus:
\[ \lambda = 2 \]

### Final Answer
The value of \( \lambda \) is \( 2 \).

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