If the adjoint of a `3 xx 3` matrix P is `[{:(,1,4,4),(,2,1,7),(,1,1,3):}]`, then the possible value(s) of the determinant of P is (are)

Options

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

Answer

To solve the problem, we need to find the possible value(s) of the determinant of the matrix \( P \) given that the adjoint of \( P \) is

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

### Step-by-Step Solution:

**Step 1: Understand the relationship between the adjoint and the determinant.**

The formula relating the determinant of a matrix and its adjoint is given by:

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

where \( n \) is the order of the matrix. For a \( 3 \times 3 \) matrix, \( n = 3 \), so:

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

**Hint for Step 1:** Remember the formula that connects the determinant of a matrix and its adjoint.

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**Step 2: Calculate the determinant of the adjoint matrix.**

We need to compute the determinant of the adjoint matrix:

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

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

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

where \( A = \begin{pmatrix} a & b & c \\ d & e & f \\ g & h & i \end{pmatrix} \).

For our matrix:

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

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. \( 1(3 - 7) = 1 \cdot (-4) = -4 \)
2. \( -4(6 - 7) = -4 \cdot (-1) = 4 \)
3. \( 4(2 - 1) = 4 \cdot 1 = 4 \)

Adding these together:

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

**Hint for Step 2:** Break down the determinant calculation into smaller parts and apply the determinant formula carefully.

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**Step 3: Relate the determinant of the adjoint to the determinant of \( P \).**

From our earlier relationship, we have:

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

Substituting the value we found:

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

**Step 4: Solve for the determinant of \( P \).**

Taking the square root of both sides gives:

\[ \text{det}(P) = \sqrt{4} = 2 \quad \text{or} \quad \text{det}(P) = -\sqrt{4} = -2. \]

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

**Final Answer: The possible values of the determinant of \( P \) are \( 2 \) and \( -2 \).**

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