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This image contains a matrix operation activity on a 2x3 matrix \( B = \begin{bmatrix} 3 & 1 & 1 \\ -1 & -3 & 1 \end{bmatrix} \), where a sequence of row and column operations are applied. The operations are described step-by-step with intermediate matrices left blank for completion.

Let's walk through the operations described:

1. **\( R_1 \leftrightarrow R_2 \)**: Interchange Row 1 and Row 2.
   \[
   B = \begin{bmatrix} -1 & -3 & 1 \\ 3 & 1 & 1 \end{bmatrix}
   \]

2. **\( C_1 \rightarrow C_1 + 2C_3 \)**: Add two times Column 3 to Column 1.
   \[
   B = \begin{bmatrix} -1 + 2 \cdot 1 & -3 & 1 \\ 3 + 2 \cdot 1 & 1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & -3 & 1 \\ 5 & 1 & 1 \end{bmatrix}
   \]

3. **\( C_2 \rightarrow C_2 - C_3 \)**: Subtract Column 3 from Column 2.
   \[
   B = \begin{bmatrix} 1 & -3 - 1 & 1 \\ 5 & 1 - 1 & 1 \end{bmatrix} = \begin{bmatrix} 1 & -4 & 1 \\ 5 & 0 & 1 \end{bmatrix}
   \]

4. **\( C_{23} \)**: This operation isn't clear from the description (could refer to interchanging or performing some operation on columns 2 and 3). Let's assume it's a column swap \( C_2 \leftrightarrow C_3 \).
   \[
   B = \begin{bmatrix} 1 & 1 & -4 \\ 5 & 1 & 0 \end{bmatrix}
   \]

5. **\( R_2 \rightarrow R_2 + 3R_1 \)**: Add three times Row 1 to Row 2.
   \[
   B = \begin{bmatrix} 1 & 1 & -4 \\ 5 + 3 \cdot 1 & 1 + 3 \cdot 1 & 0 + 3 \cdot (-4) \end{bmatrix} = \begin{bmatrix} 1 & 1 & -4 \\ 8 & 4 & -12 \end{bmatrix}
   \]

This sequence of row and column operations transforms the matrix as described.

Do you want more details on any of these steps? Let me know if you have any questions.

Here are five related questions to deepen your understanding:

1. What happens when you add a multiple of one row to another row in terms of linear transformations?
2. How do column operations differ from row operations in terms of solving systems of linear equations?
3. What is the effect of row and column swaps on the determinant of a square matrix?
4. How can Gaussian elimination be used to simplify matrices like this one?
5. What are the key differences between elementary row operations and column operations?

**Tip:** Interchanging rows or columns affects the determinant of a square matrix, so always keep track of swaps when performing these operations on a square matrix!

Activity: Apply the given row and column operations on matrix B = [ [3, 1, 1], [-1, -3, 1] ]. Operations: (i) R1 ↔ R2, (ii) C1 → C1 + 2C3, (iii) C2 → C2 - C3, (iv) C23, (v) R2 + 3R1.

This problem walks through matrix operations on a 2x3 matrix B = [ [3, 1, 1], [-1, -3, 1] ]. The task involves applying row and column operations such as row swaps, column additions, and linear combinations of rows and columns. A step-by-step solution is provided to help understand these transformations.

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