If `A=[[1, 2, 3], [3, 4, 5], [4, 6, 8]]`, then

Options

A. A is non-singular
B. `A^(-1)` does not exists
C. `A^(-1)` exists
D. `A A^(-1)=1`

Answer

To solve the problem, we need to determine the properties of the matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 4 & 6 & 8 \end{bmatrix} \). Specifically, we will find the determinant of the matrix to check if it is singular or non-singular, and based on that, we will evaluate the options provided.

### Step 1: Calculate the Determinant of Matrix A

The determinant of a 3x3 matrix \( A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \) can be calculated using the formula:

\[ \text{det}(A) = a(ei - fh) - b(di - fg) + c(dh - eg) \]

For our matrix \( A = \begin{bmatrix} 1 & 2 & 3 \\ 3 & 4 & 5 \\ 4 & 6 & 8 \end{bmatrix} \), we have:

- \( a = 1, b = 2, c = 3 \)
- \( d = 3, e = 4, f = 5 \)
- \( g = 4, h = 6, i = 8 \)

Substituting these values into the determinant formula:

\[ \text{det}(A) = 1(4 \cdot 8 - 5 \cdot 6) - 2(3 \cdot 8 - 5 \cdot 4) + 3(3 \cdot 6 - 4 \cdot 4) \]

Calculating each term:

1. \( 4 \cdot 8 = 32 \)
2. \( 5 \cdot 6 = 30 \)
3. \( 3 \cdot 8 = 24 \)
4. \( 5 \cdot 4 = 20 \)
5. \( 3 \cdot 6 = 18 \)
6. \( 4 \cdot 4 = 16 \)

Now substituting these back into the determinant calculation:

\[ \text{det}(A) = 1(32 - 30) - 2(24 - 20) + 3(18 - 16) \]

Calculating each part:

- \( 32 - 30 = 2 \)
- \( 24 - 20 = 4 \)
- \( 18 - 16 = 2 \)

Now substituting these results:

\[ \text{det}(A) = 1(2) - 2(4) + 3(2) \]
\[ = 2 - 8 + 6 \]
\[ = 2 - 8 + 6 = 0 \]

### Conclusion

Since the determinant of matrix \( A \) is \( 0 \), this indicates that the matrix is singular. Therefore, the options can be evaluated as follows:

1. **Option 1**: A is non-singular - **False**
2. **Option 2**: A inverse does not exist - **True**
3. **Option 3**: A inverse exists - **False**
4. **Option 4**: A into A inverse is equal to I - **False**

### Final Answer

The correct option is: **A inverse does not exist**.

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