Let M be a `3xx3` non-singular matrix with det(M)=`4,"If" M^(-1)"adj(adjM)=k^(2)I`, then the value of 'k' may be

Options

A. `+2`
B. 4
C. `-2`
D. `-4`

Answer

To solve the problem, we need to find the value of \( k \) given the equation \( M^{-1} \text{adj}(\text{adj} M) = k^2 I \) for a non-singular \( 3 \times 3 \) matrix \( M \) with \( \text{det}(M) = 4 \).

### Step-by-Step Solution:

1. **Understanding the Given Information:**
We know that \( M \) is a \( 3 \times 3 \) non-singular matrix, which means its determinant is not zero. We are given that \( \text{det}(M) = 4 \).

2. **Using the Formula for Inverse:**
The inverse of a matrix \( M \) can be expressed as:
\[ M^{-1} = \frac{1}{\text{det}(M)} \text{adj}(M) \]
Therefore, substituting the determinant value:
\[ M^{-1} = \frac{1}{4} \text{adj}(M) \]

3. **Using the Formula for Adjoint of Adjoint:**
For any square matrix \( M \), the adjoint of the adjoint can be given by:
\[ \text{adj}(\text{adj}(M)) = \text{det}(M)^{n-2} M \]
where \( n \) is the order of the matrix. Since \( M \) is \( 3 \times 3 \):
\[ \text{adj}(\text{adj}(M)) = \text{det}(M)^{3-2} M = \text{det}(M) \cdot M = 4M \]

4. **Substituting into the Given Equation:**
Now we substitute \( M^{-1} \) and \( \text{adj}(\text{adj}(M)) \) into the equation:
\[ M^{-1} \text{adj}(\text{adj}(M)) = \left(\frac{1}{4} \text{adj}(M)\right)(4M) \]
Simplifying this gives:
\[ M^{-1} \text{adj}(\text{adj}(M)) = \text{adj}(M) M \]

5. **Using the Property of Adjoint:**
We know that:
\[ \text{adj}(M) M = \text{det}(M) I \]
Thus:
\[ \text{adj}(M) M = 4I \]

6. **Equating to Find \( k \):**
Now we have:
\[ k^2 I = 4I \]
This implies:
\[ k^2 = 4 \]

7. **Finding the Values of \( k \):**
Taking the square root of both sides gives:
\[ k = \pm 2 \]

### Final Answer:
The value of \( k \) may be \( 2 \) or \( -2 \).