Which of the statements is sufficient to answer the question ?
Question:
Find the value of n .
Statements:
(1) `A = [ (n,0),(1,1)]`
(2) `B = [ (1,0),(-1,1)]`
(3) `A + B = O,` where O is a null matrix of `2 xx 2`

Options

A. All statements 1,2 and 3 are sufficient
B. Only statement 2 is sufficient
C. Only statement 1 is sufficient
D. Only statement 3 is sufficient

Answer

To solve the problem, we need to determine the value of \( n \) using the given statements about matrices \( A \) and \( B \). Let's go through the steps systematically.

### Step 1: Define the matrices

We are given two matrices:
- Matrix \( A \):
\[ A = \begin{pmatrix} n & 0 \\ 1 & 1 \end{pmatrix} \]
- Matrix \( B \):
\[ B = \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix} \]

### Step 2: Set up the equation for the null matrix

We need to find \( n \) such that the sum of matrices \( A \) and \( B \) equals the null matrix \( O \):
\[ A + B = O \]
Where \( O \) is the null matrix:
\[ O = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

### Step 3: Calculate \( A + B \)

Now, we will add matrices \( A \) and \( B \):
\[ A + B = \begin{pmatrix} n & 0 \\ 1 & 1 \end{pmatrix} + \begin{pmatrix} 1 & 0 \\ -1 & 1 \end{pmatrix} \]

Adding the corresponding elements:
\[ A + B = \begin{pmatrix} n + 1 & 0 + 0 \\ 1 - 1 & 1 + 1 \end{pmatrix} = \begin{pmatrix} n + 1 & 0 \\ 0 & 2 \end{pmatrix} \]

### Step 4: Set the sum equal to the null matrix

Now we set \( A + B \) equal to the null matrix \( O \):
\[ \begin{pmatrix} n + 1 & 0 \\ 0 & 2 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} \]

### Step 5: Solve the equations

From the equation above, we can derive two equations:
1. \( n + 1 = 0 \)
2. \( 2 = 0 \)

The second equation \( 2 = 0 \) is not valid, indicating that the second row does not contribute to finding \( n \). We will only focus on the first equation:
\[ n + 1 = 0 \implies n = -1 \]

### Conclusion

Thus, the value of \( n \) is:
\[ \boxed{-1} \]