If `A=[[1, 2, 3], [1, 3, 4], [3, 4, 3]]`, then `|A^(-1)|=`

Options

A. `(1)/(4)`
B. `(-1)/(4)`
C. `4`
D. `-4`

Answer

To find the determinant of the inverse of matrix \( A \), we can use the property that states:

\[ |A^{-1}| = \frac{1}{|A|} \]

First, we need to calculate the determinant of matrix \( A \).

Given:
\[ A = \begin{bmatrix} 1 & 2 & 3 \\ 1 & 3 & 4 \\ 3 & 4 & 3 \end{bmatrix} \]

### Step 1: Calculate the determinant of \( A \)

We can calculate the determinant of \( A \) using the formula for the determinant of a 3x3 matrix:

\[ |A| = a(ei - fh) - b(di - fg) + c(dh - eg) \]

Where the matrix is represented as:
\[ \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \]

For our matrix \( A \):
- \( a = 1, b = 2, c = 3 \)
- \( d = 1, e = 3, f = 4 \)
- \( g = 3, h = 4, i = 3 \)

Substituting these values into the determinant formula:

\[ |A| = 1(3 \cdot 3 - 4 \cdot 4) - 2(1 \cdot 3 - 4 \cdot 3) + 3(1 \cdot 4 - 3 \cdot 3) \]

Calculating each part:

1. \( 3 \cdot 3 - 4 \cdot 4 = 9 - 16 = -7 \)
2. \( 1 \cdot 3 - 4 \cdot 3 = 3 - 12 = -9 \)
3. \( 1 \cdot 4 - 3 \cdot 3 = 4 - 9 = -5 \)

Now substituting back into the determinant formula:

\[ |A| = 1(-7) - 2(-9) + 3(-5) \]
\[ |A| = -7 + 18 - 15 \]
\[ |A| = -4 \]

### Step 2: Calculate the determinant of \( A^{-1} \)

Using the property of determinants:

\[ |A^{-1}| = \frac{1}{|A|} = \frac{1}{-4} = -\frac{1}{4} \]

### Final Answer

Thus, the determinant of \( A^{-1} \) is:

\[ |A^{-1}| = -\frac{1}{4} \]

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