If `A=[(3,-2),(7,-5)]`, then the value of `|-3A^(2019)+A^(2020)|` is equal to

Options

A. `-14`
B. 28
C. 14
D. `2^(2019)*14`

Answer

To solve the problem, we need to find the value of \(|-3A^{2019} + A^{2020}|\) where \(A = \begin{pmatrix} 3 & -2 \\ 7 & -5 \end{pmatrix}\).

### Step-by-Step Solution:

1. **Identify the Matrix and Its Properties**:
We have the matrix \(A\):
\[ A = \begin{pmatrix} 3 & -2 \\ 7 & -5 \end{pmatrix} \]

2. **Calculate the Determinant of Matrix A**:
The determinant of a 2x2 matrix \(\begin{pmatrix} a & b \\ c & d \end{pmatrix}\) is given by \(ad - bc\).
\[ |A| = 3 \cdot (-5) - (-2) \cdot 7 = -15 + 14 = -1 \]

3. **Express the Given Expression**:
We can rewrite the expression \(|-3A^{2019} + A^{2020}|\) as:
\[ |-3A^{2019} + A^{2020}| = |A^{2019}(-3I + A)| \]
where \(I\) is the identity matrix.

4. **Factor Out \(A^{2019}\)**:
Since the determinant of a product of matrices is the product of their determinants, we have:
\[ |A^{2019}(-3I + A)| = |A^{2019}| \cdot |-3I + A| \]

5. **Calculate \(|A^{2019}|\)**:
The determinant of \(A^{n}\) is given by \(|A|^n\):
\[ |A^{2019}| = |A|^{2019} = (-1)^{2019} = -1 \]

6. **Calculate \(-3I + A\)**:
First, we find \(-3I\):
\[ -3I = \begin{pmatrix} -3 & 0 \\ 0 & -3 \end{pmatrix} \]
Now, add \(A\):
\[ -3I + A = \begin{pmatrix} -3 & 0 \\ 0 & -3 \end{pmatrix} + \begin{pmatrix} 3 & -2 \\ 7 & -5 \end{pmatrix} = \begin{pmatrix} 0 & -2 \\ 7 & -8 \end{pmatrix} \]

7. **Calculate the Determinant of \(-3I + A\)**:
\[ |-3I + A| = | \begin{pmatrix} 0 & -2 \\ 7 & -8 \end{pmatrix} | = (0)(-8) - (-2)(7) = 0 + 14 = 14 \]

8. **Combine the Results**:
Now, we combine the determinants:
\[ |-3A^{2019} + A^{2020}| = |A^{2019}| \cdot |-3I + A| = (-1) \cdot 14 = -14 \]

### Final Answer:
Thus, the value of \(|-3A^{2019} + A^{2020}|\) is \(-14\).