If `{:A=[(1,2,2),(2,3,0),(0,1,2)]and adjA=[(6,-2,-6),(-4,2,x),(y,-1,-1)]:}`,then x + y =

Options

A. 6
B. -1
C. 3
D. 1

Answer

To solve the problem, we need to find the values of \( x \) and \( y \) from the given matrices \( A \) and \( \text{adj} A \), and then compute \( x + y \).

Given:
\[ A = \begin{pmatrix} 1 & 2 & 2 \\ 2 & 3 & 0 \\ 0 & 1 & 2 \end{pmatrix} \]
\[ \text{adj} A = \begin{pmatrix} 6 & -2 & -6 \\ -4 & 2 & x \\ y & -1 & -1 \end{pmatrix} \]

### Step 1: Calculate the determinant of matrix \( A \)

The determinant of a \( 3 \times 3 \) matrix \( A \) can be calculated using the formula:
\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]
For matrix \( A \):
\[ \text{det}(A) = 1(3 \cdot 2 - 0 \cdot 1) - 2(2 \cdot 2 - 0 \cdot 0) + 2(2 \cdot 1 - 3 \cdot 0) \]
Calculating each term:
\[ = 1(6) - 2(4) + 2(2) \]
\[ = 6 - 8 + 4 = 2 \]

### Step 2: Use the property of adjoint

The property of adjoint states that:
\[ \text{adj}(A) = \text{det}(A) \cdot A^{-1} \]
From the properties of determinants, we also know that:
\[ \text{adj}(A) = \text{det}(A) \cdot \text{adj}(A) \]
This means that the sum of the elements in the adjoint matrix corresponding to the determinant can be used to find \( x \) and \( y \).

### Step 3: Relate the adjoint matrix to the determinant

Since \( \text{det}(A) = 2 \), we can use the relation:
\[ \text{adj}(A) = \text{det}(A) \cdot \text{Cofactor}(A) \]
From the given adjoint matrix, we can equate the corresponding elements:
\[ \begin{pmatrix} 6 & -2 & -6 \\ -4 & 2 & x \\ y & -1 & -1 \end{pmatrix} \]

### Step 4: Compare elements to find \( x \) and \( y \)

From the adjoint matrix:
- The element at position \( (2, 3) \) gives us \( x \).
- The element at position \( (3, 1) \) gives us \( y \).

We can directly use the values from the adjoint matrix:
- From \( \text{adj}(A) \), we see that \( x = -6 \) and \( y = 2 \).

### Step 5: Calculate \( x + y \)

Now we can find \( x + y \):
\[ x + y = -6 + 2 = -4 \]

### Final Answer
Thus, the value of \( x + y \) is:
\[ \boxed{-4} \]