If A and B are any `2xx2` matrices, then det (A+B)=0 implies:

Options

A. det A=0 and det B=0
B. det A+det `B!=0`
C. det A=0 or det B=0
D. None of these

Answer

To solve the problem, we need to analyze the implications of the determinant of the sum of two 2x2 matrices being zero. Let's denote the matrices as \( A \) and \( B \).

### Step-by-Step Solution:

1. **Understanding the Determinant Condition**:
We are given that \( \text{det}(A + B) = 0 \). This means that the matrix \( A + B \) is singular, i.e., it does not have an inverse.

2. **Exploring the Properties of Determinants**:
The determinant of a sum of matrices does not have a straightforward relationship with the determinants of the individual matrices. However, we can explore specific cases to understand the implications.

3. **Choosing Specific Matrices**:
Let's consider two specific matrices:
\[ A = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix}, \quad B = \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} \]
Now, we compute \( A + B \):
\[ A + B = \begin{pmatrix} 1 & 0 \\ 1 & 0 \end{pmatrix} + \begin{pmatrix} -1 & 0 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 1 & -1 \end{pmatrix} \]

4. **Calculating the Determinant of \( A + B \)**:
Now, we calculate the determinant:
\[ \text{det}(A + B) = \text{det}\begin{pmatrix} 0 & 0 \\ 1 & -1 \end{pmatrix} = (0)(-1) - (0)(1) = 0 \]
This confirms that \( \text{det}(A + B) = 0 \).

5. **Calculating the Determinants of \( A \) and \( B \)**:
Next, we calculate the determinants of \( A \) and \( B \):
\[ \text{det}(A) = (1)(0) - (0)(1) = 0 \]
\[ \text{det}(B) = (-1)(-1) - (0)(0) = 1 \]

6. **Analyzing the Options**:
Now we analyze the options given in the question:
- **Option 1**: \( \text{det}(A) = 0 \) and \( \text{det}(B) = 0 \) (False, since \( \text{det}(B) = 1 \))
- **Option 2**: \( \text{det}(A) + \text{det}(B) \neq 0 \) (True, since \( 0 + 1 \neq 0 \))
- **Option 3**: \( \text{det}(A) = 0 \) or \( \text{det}(B) = 0 \) (True, since \( \text{det}(A) = 0 \))
- **Option 4**: None of these (False, since Option 2 and Option 3 are true)

7. **Conclusion**:
The correct implication of \( \text{det}(A + B) = 0 \) is that at least one of the matrices \( A \) or \( B \) must be singular (i.e., have a determinant of zero). Thus, the correct answer is that \( \text{det}(A) = 0 \) or \( \text{det}(B) = 0 \).

### Final Answer:
The correct implication is that \( \text{det}(A) = 0 \) or \( \text{det}(B) = 0 \).