Given a square matrix A of order `3xx3` such that `|A|=12`, find the value of `|"A adj A"|`

Answer

To find the value of \( |A \, \text{adj} \, A| \) given that \( |A| = 12 \) for a square matrix \( A \) of order \( 3 \times 3 \), we can follow these steps:

### Step 1: Understand the relationship between a matrix and its adjugate
The adjugate of a matrix \( A \), denoted as \( \text{adj} \, A \), is related to the determinant of \( A \) by the formula:
\[ A \cdot \text{adj} \, A = |A| I \]
where \( I \) is the identity matrix of the same order as \( A \).

### Step 2: Use the property of determinants
Taking the determinant on both sides of the equation \( A \cdot \text{adj} \, A = |A| I \), we have:
\[ |A \cdot \text{adj} \, A| = | |A| I | \]

### Step 3: Apply the determinant property
Using the property of determinants, we know that:
\[ |A \cdot B| = |A| \cdot |B| \]
Thus, we can write:
\[ |A| \cdot |\text{adj} \, A| = |A|^3 \]
since the determinant of the identity matrix \( I \) of order \( n \) is \( 1 \).

### Step 4: Substitute the known value of \( |A| \)
Given that \( |A| = 12 \), we can substitute this into the equation:
\[ 12 \cdot |\text{adj} \, A| = 12^3 \]

### Step 5: Calculate \( 12^3 \)
Calculating \( 12^3 \):
\[ 12^3 = 12 \times 12 \times 12 = 1728 \]

### Step 6: Solve for \( |\text{adj} \, A| \)
Now, we can solve for \( |\text{adj} \, A| \):
\[ |\text{adj} \, A| = \frac{12^3}{12} = 12^2 = 144 \]

### Step 7: Find the value of \( |A \, \text{adj} \, A| \)
Now, we can find \( |A \, \text{adj} \, A| \):
\[ |A \, \text{adj} \, A| = |A| \cdot |\text{adj} \, A| = 12 \cdot 144 = 1728 \]

Thus, the final answer is:
\[ \boxed{1728} \]