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

Options

A. `{:[(3n,-4n),(n,-n)]:}`
B. `{:[(2+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 calculate \( X^2 \) and \( X^3 \) to observe any patterns.

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

We start by calculating \( X^2 \) which is \( X \times X \).

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

To perform the multiplication, we calculate each element:

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

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

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

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

Thus, we have:
\[ X^2 = \begin{pmatrix} 5 & -8 \\ 2 & -3 \end{pmatrix} \]

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

Next, we calculate \( X^3 \) which is \( X^2 \times X \).

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

Calculating each element:

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

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

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

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

Thus, we have:
\[ X^3 = \begin{pmatrix} 7 & -12 \\ 3 & -5 \end{pmatrix} \]

### Step 3: Identify the Pattern

Now that we have \( X^2 \) and \( X^3 \), we can look for a pattern.

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

From the calculations, we can see that the matrices are changing in a way that suggests a recurrence relation might exist.

### Conclusion

The value of \( X^n \) can be expressed in terms of its previous powers, and we can derive a general formula or continue calculating for higher powers to find a specific pattern.