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 the determinant \( |-3A^{2019} + A^{2020}| \) where \( A = \begin{pmatrix} 3 & -2 \\ 7 & -5 \end{pmatrix} \).

### Step 1: Factor out \( A^{2019} \)
We can rewrite the expression inside the determinant:
\[ -3A^{2019} + A^{2020} = A^{2019}(-3I + A) \]
where \( I \) is the identity matrix.

### Step 2: Use the property of determinants
Using the property of determinants, we have:
\[ | -3A^{2019} + A^{2020} | = | A^{2019}(-3I + A) | \]
This can be simplified using the determinant property:
\[ |A^{2019}(-3I + A)| = |A^{2019}| \cdot |-3I + A| \]

### Step 3: Calculate \( |A^{2019}| \)
The determinant of \( A^{2019} \) can be calculated as:
\[ |A^{2019}| = |A|^{2019} \]

### Step 4: Calculate \( |A| \)
Now we need to find \( |A| \):
\[ |A| = \begin{vmatrix} 3 & -2 \\ 7 & -5 \end{vmatrix} = (3)(-5) - (-2)(7) = -15 + 14 = -1 \]

Thus,
\[ |A| = -1 \]
Therefore,
\[ |A^{2019}| = (-1)^{2019} = -1 \]

### Step 5: Calculate \( |-3I + A| \)
Next, we calculate \( |-3I + A| \):
\[ -3I + A = \begin{pmatrix} -3 & 0 \\ 0 & -3 \end{pmatrix} + \begin{pmatrix} 3 & -2 \\ 7 & -5 \end{pmatrix} = \begin{pmatrix} 0 & -2 \\ 7 & -8 \end{pmatrix} \]
Now, we find the determinant:
\[ |-3I + A| = \begin{vmatrix} 0 & -2 \\ 7 & -8 \end{vmatrix} = (0)(-8) - (-2)(7) = 0 + 14 = 14 \]

### Step 6: Combine results
Now we combine the results:
\[ |-3A^{2019} + A^{2020}| = |A^{2019}| \cdot |-3I + A| = (-1) \cdot 14 = -14 \]

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

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