Let's go through each question step by step and calculate the solutions.

Question 1:

What is the trace of the matrix
$A = \begin{bmatrix} 1 & 2 & 3 & 4 \\ 5 & 6 & 7 & 8 \\ 9 & 10 & 11 & 12 \\ 13 & 14 & 15 & 16 \end{bmatrix} ?$

The trace of a matrix is the sum of the elements on its main diagonal. The main diagonal elements are the ones from the top left to the bottom right.

For matrix $A$, the elements on the main diagonal are: $1, 6, 11, 16$

So, the trace is: $\text{Trace}(A) = 1 + 6 + 11 + 16 = 34$

Thus, the answer is (c) 34.

Question 2:

If
$A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \quad \text{and} \quad B = \begin{bmatrix} 5 & 6 \\ 7 & 8 \end{bmatrix}$ what is the result of the matrix multiplication $AB$?

To calculate the matrix product $AB$, we use the formula for matrix multiplication: $(AB)_{ij} = \sum_k A_{ik} B_{kj}$

Now, let's multiply: $AB = \begin{bmatrix} (1 \cdot 5 + 2 \cdot 7) & (1 \cdot 6 + 2 \cdot 8) \\ (3 \cdot 5 + 4 \cdot 7) & (3 \cdot 6 + 4 \cdot 8) \end{bmatrix}$

Calculating each element: $AB = \begin{bmatrix} (5 + 14) & (6 + 16) \\ (15 + 28) & (18 + 32) \end{bmatrix} = \begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$

Thus, the answer is (a) $\begin{bmatrix} 19 & 22 \\ 43 & 50 \end{bmatrix}$.

Question 3:

Given matrix
$C = \begin{bmatrix} -1 & 2 \\ 3 & 0 \end{bmatrix} \quad \text{and vector} \quad d = \begin{bmatrix} 4 \\ -2 \end{bmatrix}$ compute $Cd$.

To multiply a matrix by a vector, we use the same multiplication rule as for matrix-matrix multiplication.

Let's compute $Cd$: $Cd = \begin{bmatrix} (-1 \cdot 4 + 2 \cdot (-2)) \\ (3 \cdot 4 + 0 \cdot (-2)) \end{bmatrix} = \begin{bmatrix} (-4 - 4) \\ (12 + 0) \end{bmatrix} = \begin{bmatrix} -8 \\ 12 \end{bmatrix}$

Thus, the answer is (b) $\begin{bmatrix} -8 \\ 12 \end{bmatrix}$.

Would you like to explore further details for any of these steps?

Here are five related questions to extend the discussion:

1. What is the trace of a 3x3 matrix with all diagonal elements equal to 5?
2. How would you compute the determinant of the matrix $A$ in Question 1?
3. What is the result of the matrix multiplication $BA$ instead of $AB$ for Question 2?
4. If the vector $d$ in Question 3 was changed to $\begin{bmatrix} -2 \\ 4 \end{bmatrix}$, what would $Cd$ become?
5. Can you explain the geometric interpretation of matrix-vector multiplication like in Question 3?

Tip: When multiplying matrices, ensure that the number of columns in the first matrix matches the number of rows in the second matrix, as this is a condition for valid matrix multiplication.