If the adjoint of a 3x3 matrix P is (1 4 4) (2 1 7) (1 1 3), then the possible value(s) of the determinant of P is (are)

• A) -2
• B) -1
• C) 1
• D) 2

To find the possible values of the determinant of the matrix \( P \) given its adjoint, we can use the relationship between a matrix and its adjoint. The adjoint of a matrix \( P \), denoted as \( \text{adj}(P) \), is related to the determinant of \( P \) by the formula:

Key Relationship

For any square matrix \( P \), the following holds:

• \( \text{adj}(P) = \det(P) \cdot P^{-1} \)

This means that the determinant of \( P \) can be found using the determinant of its adjoint.

Determinant of the Adjoint

The determinant of the adjoint of a \( 3 \times 3 \) matrix is given by:

• \( \det(\text{adj}(P)) = (\det(P))^2 \)

Calculating the Determinant of the Adjoint

Now, let's calculate the determinant of the given adjoint matrix:

• Adjoint matrix:

          | 1 4 4 |
          | 2 1 7 |
          | 1 1 3 |
          

Calculating the determinant:

• Using the formula for the determinant of a \( 3 \times 3 \) matrix:

det = 1(1*3 - 7*1) - 4(2*3 - 7*1) + 4(2*1 - 1*1)
det = 1(3 - 7) - 4(6 - 7) + 4(2 - 1)
det = 1(-4) - 4(-1) + 4(1)
det = -4 + 4 + 4 = 4

Finding the Determinant of P

Since we have \( \det(\text{adj}(P)) = 4 \), we can set up the equation:

• \( (\det(P))^2 = 4 \)

This gives us two possible values for \( \det(P) \):

• \( \det(P) = 2 \)
• \( \det(P) = -2 \)

Final Answer

The possible values of the determinant of \( P \) are:

• A) -2
• D) 2