If A = `[(0,1),(1,1)]` and B =` [(0,-1),(1,0)]`then which of the following is correct ?

Options

A. (A+B) . (A-B) = `A^(2) + B^(2) `
B. `(A+B) . (A-B) = A^(2) -B^(2)`
C. (A+B) . (A-B) =I
D. None of these

Answer

To solve the problem, we need to evaluate the expressions involving the matrices A and B. Let's go through the steps systematically.

### Given Matrices:
Let
\[ A = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} \]
and
\[ B = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \]

### Step 1: Calculate \( A + B \)
To find \( A + B \), we add the corresponding elements of matrices A and B.

\[ A + B = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} + \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 + 0 & 1 + (-1) \\ 1 + 1 & 1 + 0 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 2 & 1 \end{pmatrix} \]

### Step 2: Calculate \( A - B \)
Now, we calculate \( A - B \) by subtracting the corresponding elements of B from A.

\[ A - B = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} - \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 - 0 & 1 - (-1) \\ 1 - 1 & 1 - 0 \end{pmatrix} = \begin{pmatrix} 0 & 2 \\ 0 & 1 \end{pmatrix} \]

### Step 3: Calculate \( A^2 \)
Next, we compute \( A^2 \) by multiplying matrix A by itself.

\[ A^2 = A \cdot A = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} \cdot \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} = \begin{pmatrix} 0 \cdot 0 + 1 \cdot 1 & 0 \cdot 1 + 1 \cdot 1 \\ 1 \cdot 0 + 1 \cdot 1 & 1 \cdot 1 + 1 \cdot 1 \end{pmatrix} = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix} \]

### Step 4: Calculate \( B^2 \)
Now, we compute \( B^2 \) by multiplying matrix B by itself.

\[ B^2 = B \cdot B = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \cdot \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} = \begin{pmatrix} 0 \cdot 0 + (-1) \cdot 1 & 0 \cdot (-1) + (-1) \cdot 0 \\ 1 \cdot 0 + 0 \cdot 1 & 1 \cdot (-1) + 0 \cdot 0 \end{pmatrix} = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \]

### Step 5: Check the Options
Now we need to check the options given in the problem:

1. **Option 1**: \( (A + B)(A - B) = A^2 + B^2 \)
2. **Option 2**: \( (A + B)(A - B) = A^2 - B^2 \)
3. **Option 3**: \( (A + B)(A - B) = I \) (Identity matrix)
4. **Option 4**: None of these

#### Calculate \( (A + B)(A - B) \)
Now we compute \( (A + B)(A - B) \):

\[ (A + B)(A - B) = \begin{pmatrix} 0 & 0 \\ 2 & 1 \end{pmatrix} \cdot \begin{pmatrix} 0 & 2 \\ 0 & 1 \end{pmatrix} = \begin{pmatrix} 0 \cdot 0 + 0 \cdot 0 & 0 \cdot 2 + 0 \cdot 1 \\ 2 \cdot 0 + 1 \cdot 0 & 2 \cdot 2 + 1 \cdot 1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 5 \end{pmatrix} \]

#### Compare with \( A^2 + B^2 \)
Now, we compute \( A^2 + B^2 \):

\[ A^2 + B^2 = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix} + \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 1 - 1 & 1 + 0 \\ 1 + 0 & 2 - 1 \end{pmatrix} = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix} \]

#### Compare with \( A^2 - B^2 \)
Now, we compute \( A^2 - B^2 \):

\[ A^2 - B^2 = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix} - \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 1 + 1 & 1 - 0 \\ 1 - 0 & 2 + 1 \end{pmatrix} = \begin{pmatrix} 2 & 1 \\ 1 & 3 \end{pmatrix} \]

### Conclusion
Since \( (A + B)(A - B) \) does not equal \( A^2 + B^2 \) or \( A^2 - B^2 \) or the identity matrix, we conclude that the correct option is **None of these**.