To solve this, we need to construct a matrix $B$ that represents the row operations applied to the matrix $A$. The matrix $B$ is a transformation matrix that, when multiplied by $A$, will result in the modified matrix after the specified row operations. Here are the steps:

Given Row Operations

1. $2R_3 + R_2 \Rightarrow R_2$
2. $-4R_1 \Rightarrow R_1$

Row Operation Matrix Construction

These row operations can be represented as transformations in matrix form. Matrix $B$ will be a $3 \times 3$ matrix since $A$ has three rows.

1. For the first operation $-4R_1 \Rightarrow R_1$:

   • This operation modifies only the first row. We place $-4$ in the first entry of the first row of $B$, indicating the scaling of $R_1$.

2. For the second operation $2R_3 + R_2 \Rightarrow R_2$:

   • This operation adds $2 \times R_3$ to $R_2$, so we place $2$ in the entry in the second row, third column of $B$. Since $R_2$ remains in its position, we also place $1$ in the second row, second column.

Constructing Matrix $B$

From these transformations, matrix $B$ is:

$B = \begin{pmatrix} -4 & 0 & 0 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}$

This matrix $B$ will, when multiplied by $A$, perform the specified row operations to produce the transformed matrix.

Would you like more details on matrix transformations or have any further questions?

Related Questions

1. How do elementary row operations translate to transformation matrices?
2. Why does multiplying by $B$ apply these row operations to $A$?
3. How would the matrix $B$ change if we had different row operations?
4. Can row operations be used to solve systems of linear equations?
5. What are the implications of row operations on matrix determinants?

Tip

Row operations can be efficiently represented with matrices, making it easy to apply multiple transformations in sequence through matrix multiplication.