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### Question 15:
You are given matrices:
- \( R = [3 \quad 1] \)
- \( Q = \begin{bmatrix} 1 \\ 3 \end{bmatrix} \)

We need to find the correct statement about these matrices. 

Let's calculate the expressions given in the options:

#### Option A: \( R + Q \)
\[
R + Q = \begin{bmatrix} 3 \quad 1 \end{bmatrix} + \begin{bmatrix} 1 \\ 3 \end{bmatrix}
\]
Matrix addition requires matrices to have the same dimensions, but \(R\) is \(1 \times 2\) and \(Q\) is \(2 \times 1\), so this addition is not possible.

#### Option B: \( R - Q^T \)
Transpose \(Q\), so:
\[
Q^T = \begin{bmatrix} 1 & 3 \end{bmatrix}
\]
Now subtract:
\[
R - Q^T = \begin{bmatrix} 3 & 1 \end{bmatrix} - \begin{bmatrix} 1 & 3 \end{bmatrix} = \begin{bmatrix} 2 & -2 \end{bmatrix}
\]
So, Option B is correct.

Let's verify the other options.

#### Option C: \( R + Q^T \)
\[
R + Q^T = \begin{bmatrix} 3 & 1 \end{bmatrix} + \begin{bmatrix} 1 & 3 \end{bmatrix} = \begin{bmatrix} 4 & 4 \end{bmatrix}
\]
So, Option C is also correct.

#### Option D: \( R - Q^2 \)
Calculating \(Q^2\) doesn't make sense as \(Q\) is not a square matrix, so this is incorrect.

#### Option E: \( Q - R^T \)
Since \(Q\) is a \(2 \times 1\) matrix and \(R^T\) is a \(2 \times 1\) matrix, subtraction should be:
\[
Q - R^T = \begin{bmatrix} 1 \\ 3 \end{bmatrix} - \begin{bmatrix} 3 \\ 1 \end{bmatrix} = \begin{bmatrix} -2 \\ 2 \end{bmatrix}
\]
So, this is incorrect.

The correct answers are **B and C**.

### Question 16:
Matrix \( D = \begin{bmatrix} -2 & 6 \\ 1 & 4 \end{bmatrix} \)

We need to compute \( D^2 \):
\[
D^2 = D \times D = \begin{bmatrix} -2 & 6 \\ 1 & 4 \end{bmatrix} \times \begin{bmatrix} -2 & 6 \\ 1 & 4 \end{bmatrix}
\]

Multiplying these two matrices:
\[
D^2 = \begin{bmatrix} (-2 \times -2) + (6 \times 1) & (-2 \times 6) + (6 \times 4) \\ (1 \times -2) + (4 \times 1) & (1 \times 6) + (4 \times 4) \end{bmatrix}
\]
\[
= \begin{bmatrix} 4 + 6 & -12 + 24 \\ -2 + 4 & 6 + 16 \end{bmatrix}
= \begin{bmatrix} 10 & 12 \\ 2 & 22 \end{bmatrix}
\]

So the correct answer is **D**.

Would you like to continue with the remaining questions, or do you have any specific one you'd like to focus on?

Solve the matrix operations and find the correct answers for each question based on the given matrices in questions 15-19.

Solve matrix operation problems covering addition, subtraction, multiplication, and transposition. These exercises help reinforce key matrix manipulation techniques for high school students. Suitable for Grades 10-12.

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