If A and B are square matrices of order 3, then

Options

A. ` adj (AB) =adjA + adj B `
B. `(A+B) =A^(-1) +B^(-1) `
C. ` AB= O rArr | A| =0 or [B] =0`
D. ` AB = OrArr |A| =0 and |B| =0`

Answer

To solve the problem regarding square matrices A and B of order 3, we will analyze the given statements step by step.

### Step-by-Step Solution:

1. **Understanding the Properties of Matrices**:
- A square matrix of order 3 means it has 3 rows and 3 columns.
- The adjoint of a matrix is a specific matrix derived from the original matrix, and it has certain properties related to determinants.

2. **Evaluating the First Statement**:
- The first statement claims that the adjoint of the product of two matrices (AB) is equal to the sum of their adjoints:
\[ \text{adj}(AB) = \text{adj}(A) + \text{adj}(B) \]
- This statement is incorrect. The correct property is:
\[ \text{adj}(AB) = \text{adj}(B) \cdot \text{adj}(A) \]
- Therefore, the first statement is false.

**Hint**: Remember that the adjoint of a product of matrices is the product of their adjoints in reverse order.

3. **Evaluating the Second Statement**:
- The second statement claims:
\[ A + B = A^{-1} + B^{-1} \]
- This statement is also incorrect. The addition of matrices does not relate to the addition of their inverses in this manner.

**Hint**: The addition of matrices follows the commutative property, but it does not relate to their inverses.

4. **Evaluating the Third Statement**:
- The third statement claims:
\[ AB = 0 \quad \text{if} \quad \det(A) = 0 \quad \text{or} \quad \det(B) = 0 \]
- This statement is true. If either matrix A or matrix B is singular (i.e., has a determinant of zero), then their product will also be the zero matrix.

**Hint**: A matrix is singular if its determinant is zero, which means it does not have an inverse.

5. **Evaluating the Fourth Statement**:
- The fourth statement claims:
\[ AB = 0 \quad \text{if and only if} \quad \det(A) = 0 \quad \text{and} \quad \det(B) = 0 \]
- This statement is false. The product AB can be zero if either A or B is singular, not necessarily both.

**Hint**: The "if and only if" condition is stronger than just "if", so be careful with logical implications.

### Conclusion:
- The correct answer is that the third statement is true: \( AB = 0 \) if \( \det(A) = 0 \) or \( \det(B) = 0 \).