The inverse of `[(1,2,4),(3,-19,7),(2,4,8)]` is

Options

A. `[(1,1,1),(3,9,7),(2,1,8)]`
B. `[(1,1,1),(19,7,8),(2,1,3)]`
C. `[(3,1,4),(1,2,3),(4,5,8)]`
D. does not exist

Answer

To find the inverse of the matrix \( A = \begin{pmatrix} 1 & 2 & 4 \\ 3 & -19 & 7 \\ 2 & 4 & 8 \end{pmatrix} \), we will follow these steps:

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

The determinant of a \( 3 \times 3 \) matrix is calculated using the formula:

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

For our matrix \( A \):

\[ A = \begin{pmatrix} 1 & 2 & 4 \\ 3 & -19 & 7 \\ 2 & 4 & 8 \end{pmatrix} \]

We can assign:
- \( a = 1, b = 2, c = 4 \)
- \( d = 3, e = -19, f = 7 \)
- \( g = 2, h = 4, i = 8 \)

Now we can calculate the determinant:

\[ \text{det}(A) = 1((-19 \cdot 8) - (7 \cdot 4)) - 2((3 \cdot 8) - (7 \cdot 2)) + 4((3 \cdot 4) - (-19 \cdot 2)) \]

Calculating each term:

1. \( -19 \cdot 8 = -152 \)
2. \( 7 \cdot 4 = 28 \)
3. \( 3 \cdot 8 = 24 \)
4. \( 7 \cdot 2 = 14 \)
5. \( 3 \cdot 4 = 12 \)
6. \( -19 \cdot 2 = -38 \)

Substituting back into the determinant formula:

\[ \text{det}(A) = 1(-152 - 28) - 2(24 - 14) + 4(12 + 38) \]
\[ = 1(-180) - 2(10) + 4(50) \]
\[ = -180 - 20 + 200 \]
\[ = 0 \]

### Conclusion

Since the determinant of matrix \( A \) is \( 0 \), the inverse of matrix \( A \) does not exist.

### Final Answer

The inverse of the matrix \( A \) does not exist.

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