Let's break down the exercise step by step.

Matrices Given:

• $A = \begin{pmatrix} -1 & 1 & 3 \\ 2 & 1 & 0 \end{pmatrix}$
• $B = \begin{pmatrix} -1 & 0 & 1 \\ 1 & 2 & 3 \end{pmatrix}$
• $C = \begin{pmatrix} 0 & 1 \\ 1 & 1 \end{pmatrix}$

Task (1):

We are asked to compute the following expressions, but only those that make sense (i.e., valid matrix multiplications):

• $ABC$
• $AB^T - 2C$
• $ATB + C$
• $BA^T$

We need to check the dimensionality of the matrices for valid multiplications.

1. $ABC$:

   • $A$ is a $2 \times 3$ matrix.
   • $B$ is a $2 \times 3$ matrix.
   • $C$ is a $2 \times 2$ matrix.
   • Matrix multiplication requires matching the inner dimensions, and here $ABC$ isn't possible because the inner dimensions don’t match properly between matrices $A$, $B$, and $C$.

2. $AB^T - 2C$:

   • $B^T$ is the transpose of $B$, so it becomes a $3 \times 2$ matrix.
   • $A$ is $2 \times 3$, and $B^T$ is $3 \times 2$. So, $AB^T$ is a valid multiplication resulting in a $2 \times 2$ matrix.
   • Subtracting $2C$ (which is a $2 \times 2$ matrix) from $AB^T$ is also valid.
   • Thus, $AB^T - 2C$ is valid and can be computed.

3. $ATB + C$:

   • This involves multiplying three matrices, where $T$ is likely the transpose of $A$. So $A^T$ is $3 \times 2$.
   • $B$ is $2 \times 3$.
   • $A^T B$ results in a $3 \times 3$ matrix.
   • Adding this to $C$, which is a $2 \times 2$ matrix, isn’t valid due to dimension mismatch. So this operation does not make sense.

4. $BA^T$:

   • $B$ is $2 \times 3$, and $A^T$ is $3 \times 2$.
   • The product $BA^T$ results in a valid $2 \times 2$ matrix.
   • Thus, this operation is valid.

Task (2):

Now we are asked to compute the powers of matrix $C$ and identify a pattern.

We need to compute:

• $C^2$
• $C^3$
• $C^4$
• $C^5$
• Identify the pattern and compute $C^{12}$.

Let's compute these now.Here are the computed powers of matrix $C$:

• $C^2 = \begin{pmatrix} 1 & 1 \\ 1 & 2 \end{pmatrix}$
• $C^3 = \begin{pmatrix} 1 & 2 \\ 2 & 3 \end{pmatrix}$
• $C^4 = \begin{pmatrix} 2 & 3 \\ 3 & 5 \end{pmatrix}$
• $C^5 = \begin{pmatrix} 3 & 5 \\ 5 & 8 \end{pmatrix}$

Pattern:

The entries of the matrices seem to follow the Fibonacci sequence:

• For $C^n$, the matrix entries appear to follow the pattern of successive Fibonacci numbers. For instance:
  • Top left is Fibonacci($n$).
  • Top right and bottom left are Fibonacci($n+1$).
  • Bottom right is Fibonacci($n+2$).

Using this pattern, we computed:

• $C^{12} = \begin{pmatrix} 89 & 144 \\ 144 & 233 \end{pmatrix}$, which continues the Fibonacci sequence.

Would you like more details or have any further questions? Here are 5 related questions you might consider:

1. Can you derive the general form for $C^n$ using Fibonacci numbers?
2. How does the Fibonacci sequence arise from matrix exponentiation in this case?
3. Could this pattern be generalized for other matrices similar to $C$?
4. How does matrix multiplication link to linear recurrences like the Fibonacci sequence?
5. What happens when you raise other 2x2 matrices to successive powers?

Tip: Matrix exponentiation is a powerful tool in solving linear recurrence relations such as the Fibonacci sequence and can provide efficient algorithms for computing large terms.