If A is a non - singular matrix of order 3, then which of the following is not true ?

Options

A. `|"adj A"|=|A^(2)|`
B. `(A^(-1))^(-1)=A`
C. If AB = AC, then `B!=C` where B and C are square matrices of order 3
D. `(AB)^(-1)=B^(-1)A^(-1)`, where B is a non - singular matrix of order 3

Answer

To solve the question regarding the properties of a non-singular matrix of order 3, we need to evaluate the given statements and identify which one is not true.

### Step-by-Step Solution:

1. **Understanding Non-Singular Matrix**:
A non-singular matrix is defined as a matrix whose determinant is not equal to zero. For a matrix \( A \) of order 3, this means \( \text{det}(A) \neq 0 \).

**Hint**: Remember that a non-singular matrix has an inverse, which is a key property.

2. **Evaluating the Options**:
We will analyze each option to determine if it is true or not for a non-singular matrix.

- **Option 1**: The adjugate of \( A \) (denoted as \( \text{adj}(A) \)) is given by the formula \( \text{adj}(A) = \text{det}(A) \cdot A^{-1} \). Since \( A \) is non-singular, this statement is true.

**Hint**: Recall the relationship between the adjugate and the determinant.

- **Option 2**: The property \( A \cdot A^{-1} = I \) (where \( I \) is the identity matrix) holds true for non-singular matrices. Therefore, this statement is also true.

**Hint**: The identity matrix is a fundamental concept when dealing with inverses.

- **Option 3**: The statement \( AB = AC \) implies \( B = C \) is not necessarily true for matrices. This only holds if \( A \) is non-singular. If \( A \) is singular, this statement can fail. Therefore, this option is not true.

**Hint**: Consider the implications of multiplying both sides by the inverse of \( A \).

- **Option 4**: The property \( AB^{-1} = B^{-1}A^{-1} \) is a valid property for non-singular matrices. Thus, this statement is true.

**Hint**: This property is derived from the rules of matrix multiplication and inverses.

3. **Conclusion**:
After analyzing all options, we find that the statement in **Option 3** is not true for a non-singular matrix. Therefore, the answer to the question is:

**Answer**: Option 3 is not true.