If `A=[(2,1),(7,5)]`, find `|A(adjA)|`

Answer

To solve the problem of finding \( |A(\text{adj} A)| \) for the matrix \( A = \begin{pmatrix} 2 & 1 \\ 7 & 5 \end{pmatrix} \), we will follow these steps:

### Step 1: Calculate the Determinant of Matrix A
The determinant of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by the formula:
\[ |A| = ad - bc \]
For our matrix \( A \):
- \( a = 2 \)
- \( b = 1 \)
- \( c = 7 \)
- \( d = 5 \)

Calculating the determinant:
\[ |A| = (2)(5) - (1)(7) = 10 - 7 = 3 \]

### Step 2: Find the Adjoint of Matrix A
The adjoint of a 2x2 matrix \( A = \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[ \text{adj} A = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]
For our matrix \( A \):
- \( d = 5 \)
- \( -b = -1 \)
- \( -c = -7 \)
- \( a = 2 \)

Thus, the adjoint of \( A \) is:
\[ \text{adj} A = \begin{pmatrix} 5 & -1 \\ -7 & 2 \end{pmatrix} \]

### Step 3: Calculate the Product \( A \cdot \text{adj} A \)
Now we will multiply \( A \) by \( \text{adj} A \):
\[ A \cdot \text{adj} A = \begin{pmatrix} 2 & 1 \\ 7 & 5 \end{pmatrix} \cdot \begin{pmatrix} 5 & -1 \\ -7 & 2 \end{pmatrix} \]

Calculating the elements:
1. First row, first column:
\[ 2 \cdot 5 + 1 \cdot (-7) = 10 - 7 = 3 \]
2. First row, second column:
\[ 2 \cdot (-1) + 1 \cdot 2 = -2 + 2 = 0 \]
3. Second row, first column:
\[ 7 \cdot 5 + 5 \cdot (-7) = 35 - 35 = 0 \]
4. Second row, second column:
\[ 7 \cdot (-1) + 5 \cdot 2 = -7 + 10 = 3 \]

Thus, we have:
\[ A \cdot \text{adj} A = \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} \]

### Step 4: Calculate the Determinant of \( A \cdot \text{adj} A \)
Now we calculate the determinant of the resulting matrix:
\[ |A \cdot \text{adj} A| = | \begin{pmatrix} 3 & 0 \\ 0 & 3 \end{pmatrix} | = (3)(3) - (0)(0) = 9 - 0 = 9 \]

### Final Answer
Thus, the value of \( |A(\text{adj} A)| \) is:
\[ \boxed{9} \]