If `{:X=[(3,-4),(1,-1)]:}`, the value of `X^n` is equal to

Options

A. `[(3n,-4n),(n,-n)]`
B. `[(2n+n,5-n),(n,-n)]`
C. `[(3^(n),(-4)^(n)),(1^(n),(-1)^(n))]`
D. None of these

Answer

To find the value of \( X^n \) for the matrix

\[ X = \begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix}, \]

we will first calculate \( X^2 \) and \( X^3 \) to identify any patterns.

### Step 1: Calculate \( X^2 \)

To find \( X^2 \), we multiply \( X \) by itself:

\[ X^2 = X \cdot X = \begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix} \cdot \begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix}. \]

Calculating the elements of \( X^2 \):

- First row, first column:
\[ 3 \cdot 3 + (-4) \cdot 1 = 9 - 4 = 5. \]

- First row, second column:
\[ 3 \cdot (-4) + (-4) \cdot (-1) = -12 + 4 = -8. \]

- Second row, first column:
\[ 1 \cdot 3 + (-1) \cdot 1 = 3 - 1 = 2. \]

- Second row, second column:
\[ 1 \cdot (-4) + (-1) \cdot (-1) = -4 + 1 = -3. \]

Thus, we have:

\[ X^2 = \begin{pmatrix} 5 & -8 \\ 2 & -3 \end{pmatrix}. \]

### Step 2: Calculate \( X^3 \)

Now, we calculate \( X^3 \) by multiplying \( X^2 \) by \( X \):

\[ X^3 = X^2 \cdot X = \begin{pmatrix} 5 & -8 \\ 2 & -3 \end{pmatrix} \cdot \begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix}. \]

Calculating the elements of \( X^3 \):

- First row, first column:
\[ 5 \cdot 3 + (-8) \cdot 1 = 15 - 8 = 7. \]

- First row, second column:
\[ 5 \cdot (-4) + (-8) \cdot (-1) = -20 + 8 = -12. \]

- Second row, first column:
\[ 2 \cdot 3 + (-3) \cdot 1 = 6 - 3 = 3. \]

- Second row, second column:
\[ 2 \cdot (-4) + (-3) \cdot (-1) = -8 + 3 = -5. \]

Thus, we have:

\[ X^3 = \begin{pmatrix} 7 & -12 \\ 3 & -5 \end{pmatrix}. \]

### Step 3: Identify a Pattern

From our calculations, we have:

- \( X^1 = \begin{pmatrix} 3 & -4 \\ 1 & -1 \end{pmatrix} \)
- \( X^2 = \begin{pmatrix} 5 & -8 \\ 2 & -3 \end{pmatrix} \)
- \( X^3 = \begin{pmatrix} 7 & -12 \\ 3 & -5 \end{pmatrix} \)

There does not appear to be a simple pattern or formula that can be derived from these results.

### Conclusion

Since we cannot express \( X^n \) in a simple closed form based on the calculations we performed, the correct option is that there is no general formula for \( X^n \) based on the options provided.