Let `A=[(2,3),(0,-1)]` , the value of `det(A^4)-det(A^10-adj(2A)^10))`

Answer

To solve the problem, we need to calculate the expression \( \text{det}(A^4) - \text{det}(A^{10} - \text{adj}(2A)^{10}) \).

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

Given:
\[ A = \begin{pmatrix} 2 & 3 \\ 0 & -1 \end{pmatrix} \]

The determinant of a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is calculated as:
\[ \text{det}(A) = ad - bc \]

For matrix \( A \):
\[ \text{det}(A) = (2)(-1) - (3)(0) = -2 - 0 = -2 \]

### Step 2: Calculate \( \text{det}(A^4) \)

Using the property of determinants, \( \text{det}(A^n) = (\text{det}(A))^n \):
\[ \text{det}(A^4) = (\text{det}(A))^4 = (-2)^4 = 16 \]

### Step 3: Calculate \( \text{det}(A^{10}) \)

Similarly:
\[ \text{det}(A^{10}) = (\text{det}(A))^{10} = (-2)^{10} = 1024 \]

### Step 4: Calculate \( 2A \)

Now, we calculate \( 2A \):
\[ 2A = 2 \cdot \begin{pmatrix} 2 & 3 \\ 0 & -1 \end{pmatrix} = \begin{pmatrix} 4 & 6 \\ 0 & -2 \end{pmatrix} \]

### Step 5: Calculate \( \text{adj}(2A) \)

The adjugate of a \( 2 \times 2 \) matrix \( \begin{pmatrix} a & b \\ c & d \end{pmatrix} \) is given by:
\[ \text{adj}(A) = \begin{pmatrix} d & -b \\ -c & a \end{pmatrix} \]

For \( 2A \):
\[ \text{adj}(2A) = \begin{pmatrix} -2 & -6 \\ 0 & 4 \end{pmatrix} \]

### Step 6: Calculate \( \text{det}(\text{adj}(2A)) \)

Using the property \( \text{det}(\text{adj}(A)) = (\text{det}(A))^{n-1} \) for a \( n \times n \) matrix:
\[ \text{det}(2A) = 2^2 \cdot \text{det}(A) = 4 \cdot (-2) = -8 \]
\[ \text{det}(\text{adj}(2A)) = (-8)^{1} = -8 \]

### Step 7: Calculate \( \text{det}(\text{adj}(2A)^{10}) \)

Using the property of determinants:
\[ \text{det}(\text{adj}(2A)^{10}) = (\text{det}(\text{adj}(2A)))^{10} = (-8)^{10} = 1073741824 \]

### Step 8: Calculate \( \text{det}(A^{10} - \text{adj}(2A)^{10}) \)

Since \( \text{adj}(2A)^{10} \) is a scalar multiple of the identity matrix (as it is a \( 2 \times 2 \) matrix raised to a power), we can express it as:
\[ \text{det}(A^{10} - \text{adj}(2A)^{10}) = \text{det}(1024 - 1073741824) \]
This simplifies to:
\[ \text{det}(-1073740800) = (-1073740800)^2 = 1152921504606846976000 \]

### Step 9: Final Calculation

Now, we can substitute back into our original expression:
\[ \text{det}(A^4) - \text{det}(A^{10} - \text{adj}(2A)^{10}) = 16 - 1152921504606846976000 \]

This results in:
\[ \text{det}(A^4) - \text{det}(A^{10} - \text{adj}(2A)^{10}) = -1152921504606846975984 \]

### Final Answer

Thus, the final value is:
\[ \text{det}(A^4) - \text{det}(A^{10} - \text{adj}(2A)^{10}) = -1152921504606846975984 \]