Let `A=[(-1,2,-3),(-2,0,3),(3,-3,1)]` be a mstrix, then `|A|adj(A^(-1))` is equal to

Options

A. `O_(3xx3)`
B. `I_(3)`
C. `[{:(,-1,2,-3),(,-2,0,3),(,3,-3,1):}]`
D. `[{:(,3,-3,1),(,3,0,-2),(,-1,2,-3):}]`

Answer

To solve the problem, we need to find the value of \( |A| \cdot \text{adj}(A^{-1}) \) where \( A = \begin{pmatrix} -1 & 2 & -3 \\ -2 & 0 & 3 \\ 3 & -3 & 1 \end{pmatrix} \).

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

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

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

For our matrix \( A \):

\[ |A| = -1 \cdot (0 \cdot 1 - 3 \cdot (-3)) - 2 \cdot (-2 \cdot 1 - 3 \cdot 3) - 3 \cdot (-2 \cdot (-3) - 0 \cdot 3) \]

Calculating each term:

1. First term:
\[ -1 \cdot (0 + 9) = -9 \]

2. Second term:
\[ -2 \cdot (2 - 9) = -2 \cdot (-7) = 14 \]

3. Third term:
\[ -3 \cdot (6 - 0) = -18 \]

Now, combine these results:

\[ |A| = -9 + 14 - 18 = -13 \]

### Step 2: Find the Adjoint of A

The adjoint of a matrix \( A \) is the transpose of the cofactor matrix. For the matrix \( A \), we need to find the cofactors of each element.

Calculating the cofactors:

- For \( a_{11} = -1 \):
\[ C_{11} = | \begin{pmatrix} 0 & 3 \\ -3 & 1 \end{pmatrix} | = (0 \cdot 1 - 3 \cdot (-3)) = 9 \quad \Rightarrow \quad \text{Cofactor} = 9 \]

- For \( a_{12} = 2 \):
\[ C_{12} = -| \begin{pmatrix} -2 & 3 \\ 3 & 1 \end{pmatrix} | = -((-2 \cdot 1 - 3 \cdot 3)) = -(-2 - 9) = 11 \quad \Rightarrow \quad \text{Cofactor} = -11 \]

- For \( a_{13} = -3 \):
\[ C_{13} = | \begin{pmatrix} -2 & 0 \\ 3 & -3 \end{pmatrix} | = (-2 \cdot -3 - 0 \cdot 3) = 6 \quad \Rightarrow \quad \text{Cofactor} = 6 \]

Continuing this process for all elements, we find the cofactor matrix and then transpose it to find the adjoint \( \text{adj}(A) \).

### Step 3: Find \( \text{adj}(A^{-1}) \)

Using the property of adjoints, we know:

\[ \text{adj}(A^{-1}) = \frac{1}{|A|} \text{adj}(A) \]

### Step 4: Combine Results

Now we can calculate:

\[ |A| \cdot \text{adj}(A^{-1}) = |A| \cdot \left(\frac{1}{|A|} \text{adj}(A)\right) = \text{adj}(A) \]

Since we have already calculated \( |A| = -13 \), the final result simplifies to \( \text{adj}(A) \).

### Final Answer

Thus, the value of \( |A| \cdot \text{adj}(A^{-1}) \) is equal to \( \text{adj}(A) \).