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

Options

A. -2
B. -1
C. 1
D. 2

Explanation

To find the possible values of the determinant of the matrix \( P \) given that the adjoint of \( P \) is

\[ \text{adj}(P) = \begin{pmatrix} 1 & 4 & 4 \\ 2 & 1 & 7 \\ 1 & 1 & 3 \end{pmatrix} \]

we will follow these steps:

### Step 1: Calculate the Determinant of the Adjoint Matrix

We need to find the determinant of the adjoint matrix \( \text{adj}(P) \). The determinant of a \( 3 \times 3 \) matrix

\[ \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \]

is calculated using the formula:

\[ \text{det} = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix:

\[ \begin{pmatrix} 1 & 4 & 4 \\ 2 & 1 & 7 \\ 1 & 1 & 3 \end{pmatrix} \]

we have \( a = 1, b = 4, c = 4, d = 2, e = 1, f = 7, g = 1, h = 1, i = 3 \).

Calculating the determinant:

\[ \text{det}(\text{adj}(P)) = 1(1 \cdot 3 - 7 \cdot 1) - 4(2 \cdot 3 - 7 \cdot 1) + 4(2 \cdot 1 - 1 \cdot 1) \]

Calculating each term:

1. First term: \( 1(3 - 7) = 1(-4) = -4 \)
2. Second term: \( -4(6 - 7) = -4(-1) = 4 \)
3. Third term: \( 4(2 - 1) = 4(1) = 4 \)

Now summing these values:

\[ \text{det}(\text{adj}(P)) = -4 + 4 + 4 = 4 \]

### Step 2: Use the Property of Determinants

We know that for a \( n \times n \) matrix \( P \):

\[ \text{det}(\text{adj}(P)) = (\text{det}(P))^{n-1} \]

For our case, \( n = 3 \):

\[ \text{det}(\text{adj}(P)) = (\text{det}(P))^{3-1} = (\text{det}(P))^2 \]

Substituting the value we found:

\[ 4 = (\text{det}(P))^2 \]

### Step 3: Solve for the Determinant of \( P \)

Taking the square root of both sides:

\[ \text{det}(P) = \pm \sqrt{4} = \pm 2 \]

### Conclusion

The possible values of the determinant of \( P \) are \( 2 \) and \( -2 \).

Thus, the answer is:

**Option A: -2 and Option D: 2 are correct.**