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For the above data give step by step calculations

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.

1. What is the trace of the matrix A = [ [1, 2, 3, 4], [5, 6, 7, 8], [9, 10, 11, 12], [13, 14, 15, 16] ]?
2. If A = [ [1, 2], [3, 4] ] and B = [ [5, 6], [7, 8] ], what is the result of the matrix multiplication AB?
3. Given matrix C = [ [-1, 2], [3, 0] ] and vector d = [4, -2], compute Cd.

This problem explores key concepts in linear algebra, including how to calculate the trace of a matrix, perform matrix multiplication, and multiply a matrix by a vector. Detailed step-by-step solutions provided.

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