## M2 - Row operations as matrix multiplication

#### Example 1 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_1 \to R_1 + -4R_2$$.
2. Give a $$4 \times 4$$ matrix $$N$$ that may be used to perform the row operation $$R_1 \to -5R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_1 \to R_1 + -4R_2$$ and then $$R_1 \to -5R_1$$ to $$A$$ (note the order).

1. $$C= \left[\begin{array}{cccc} 1 & -4 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$N= \left[\begin{array}{cccc} -5 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$NCA$$

#### Example 2 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$M$$ that may be used to perform the row operation $$R_3 \to R_3 + 4R_2$$.
2. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_1 \to -4R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_1 \to -4R_1$$ and then $$R_3 \to R_3 + 4R_2$$ to $$A$$ (note the order).

1. $$M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 4 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$C= \left[\begin{array}{cccc} -4 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$MCA$$

#### Example 3 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$P$$ that may be used to perform the row operation $$R_2 \to R_2 + -4R_4$$.
2. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_4 \leftrightarrow R_2$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_2 \to R_2 + -4R_4$$ and then $$R_4 \leftrightarrow R_2$$ to $$A$$ (note the order).

1. $$P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -4 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right]$$
3. $$QPA$$

#### Example 4 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$P$$ that may be used to perform the row operation $$R_1 \to R_1 + 5R_4$$.
2. Give a $$4 \times 4$$ matrix $$B$$ that may be used to perform the row operation $$R_2 \to -5R_2$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_2 \to -5R_2$$ and then $$R_1 \to R_1 + 5R_4$$ to $$A$$ (note the order).

1. $$P= \left[\begin{array}{cccc} 1 & 0 & 0 & 5 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & -5 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$PBA$$

#### Example 5 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_3 \to R_3 + 4R_1$$.
2. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_1 \to 5R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_3 \to R_3 + 4R_1$$ and then $$R_1 \to 5R_1$$ to $$A$$ (note the order).

1. $$C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 4 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$Q= \left[\begin{array}{cccc} 5 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$QCA$$

#### Example 6 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$N$$ that may be used to perform the row operation $$R_3 \to R_3 + -4R_2$$.
2. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_1 \to 5R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_1 \to 5R_1$$ and then $$R_3 \to R_3 + -4R_2$$ to $$A$$ (note the order).

1. $$N= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & -4 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$C= \left[\begin{array}{cccc} 5 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$NCA$$

#### Example 7 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_4 \leftrightarrow R_1$$.
2. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_2 \to 5R_2$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_4 \leftrightarrow R_1$$ and then $$R_2 \to 5R_2$$ to $$A$$ (note the order).

1. $$C= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right]$$
2. $$Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 5 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$QCA$$

#### Example 8 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_4 \leftrightarrow R_1$$.
2. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_4 \to 2R_4$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_4 \leftrightarrow R_1$$ and then $$R_4 \to 2R_4$$ to $$A$$ (note the order).

1. $$Q= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right]$$
2. $$C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right]$$
3. $$CQA$$

#### Example 9 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_2 \to R_2 + 4R_4$$.
2. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_4 \to -5R_4$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_4 \to -5R_4$$ and then $$R_2 \to R_2 + 4R_4$$ to $$A$$ (note the order).

1. $$Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 4 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -5 \end{array}\right]$$
3. $$QCA$$

#### Example 10 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$P$$ that may be used to perform the row operation $$R_1 \to R_1 + 3R_2$$.
2. Give a $$4 \times 4$$ matrix $$B$$ that may be used to perform the row operation $$R_3 \to 3R_3$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_3 \to 3R_3$$ and then $$R_1 \to R_1 + 3R_2$$ to $$A$$ (note the order).

1. $$P= \left[\begin{array}{cccc} 1 & 3 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$PBA$$

#### Example 11 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$B$$ that may be used to perform the row operation $$R_1 \to 4R_1$$.
2. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_2 \leftrightarrow R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_1 \to 4R_1$$ and then $$R_2 \leftrightarrow R_1$$ to $$A$$ (note the order).

1. $$B= \left[\begin{array}{cccc} 4 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$C= \left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$CBA$$

#### Example 12 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$B$$ that may be used to perform the row operation $$R_4 \to R_4 + -3R_2$$.
2. Give a $$4 \times 4$$ matrix $$P$$ that may be used to perform the row operation $$R_4 \to 4R_4$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_4 \to R_4 + -3R_2$$ and then $$R_4 \to 4R_4$$ to $$A$$ (note the order).

1. $$B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -3 & 0 & 1 \end{array}\right]$$
2. $$P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right]$$
3. $$PBA$$

#### Example 13 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$B$$ that may be used to perform the row operation $$R_1 \to R_1 + -5R_3$$.
2. Give a $$4 \times 4$$ matrix $$M$$ that may be used to perform the row operation $$R_1 \to -3R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_1 \to -3R_1$$ and then $$R_1 \to R_1 + -5R_3$$ to $$A$$ (note the order).

1. $$B= \left[\begin{array}{cccc} 1 & 0 & -5 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$M= \left[\begin{array}{cccc} -3 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$BMA$$

#### Example 14 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_3 \leftrightarrow R_4$$.
2. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_3 \to 4R_3$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_3 \leftrightarrow R_4$$ and then $$R_3 \to 4R_3$$ to $$A$$ (note the order).

1. $$Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right]$$
2. $$C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$CQA$$

#### Example 15 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$M$$ that may be used to perform the row operation $$R_3 \to R_3 + -4R_2$$.
2. Give a $$4 \times 4$$ matrix $$P$$ that may be used to perform the row operation $$R_3 \to 5R_3$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_3 \to 5R_3$$ and then $$R_3 \to R_3 + -4R_2$$ to $$A$$ (note the order).

1. $$M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & -4 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$MPA$$

#### Example 16 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_4 \leftrightarrow R_3$$.
2. Give a $$4 \times 4$$ matrix $$N$$ that may be used to perform the row operation $$R_2 \to R_2 + -4R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_2 \to R_2 + -4R_1$$ and then $$R_4 \leftrightarrow R_3$$ to $$A$$ (note the order).

1. $$Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right]$$
2. $$N= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ -4 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$QNA$$

#### Example 17 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$P$$ that may be used to perform the row operation $$R_4 \leftrightarrow R_3$$.
2. Give a $$4 \times 4$$ matrix $$B$$ that may be used to perform the row operation $$R_4 \to R_4 + -2R_3$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_4 \leftrightarrow R_3$$ and then $$R_4 \to R_4 + -2R_3$$ to $$A$$ (note the order).

1. $$P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right]$$
2. $$B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & -2 & 1 \end{array}\right]$$
3. $$BPA$$

#### Example 18 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_4 \to R_4 + 2R_3$$.
2. Give a $$4 \times 4$$ matrix $$N$$ that may be used to perform the row operation $$R_2 \leftrightarrow R_4$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_4 \to R_4 + 2R_3$$ and then $$R_2 \leftrightarrow R_4$$ to $$A$$ (note the order).

1. $$Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 2 & 1 \end{array}\right]$$
2. $$N= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right]$$
3. $$NQA$$

#### Example 19 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$M$$ that may be used to perform the row operation $$R_2 \to R_2 + 4R_3$$.
2. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_3 \to 5R_3$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_2 \to R_2 + 4R_3$$ and then $$R_3 \to 5R_3$$ to $$A$$ (note the order).

1. $$M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 4 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$QMA$$

#### Example 20 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$P$$ that may be used to perform the row operation $$R_4 \to R_4 + 4R_1$$.
2. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_4 \leftrightarrow R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_4 \to R_4 + 4R_1$$ and then $$R_4 \leftrightarrow R_1$$ to $$A$$ (note the order).

1. $$P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 4 & 0 & 0 & 1 \end{array}\right]$$
2. $$Q= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right]$$
3. $$QPA$$

#### Example 21 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$M$$ that may be used to perform the row operation $$R_1 \to R_1 + 5R_2$$.
2. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_2 \leftrightarrow R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_2 \leftrightarrow R_1$$ and then $$R_1 \to R_1 + 5R_2$$ to $$A$$ (note the order).

1. $$M= \left[\begin{array}{cccc} 1 & 5 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$Q= \left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$MQA$$

#### Example 22 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_1 \to 2R_1$$.
2. Give a $$4 \times 4$$ matrix $$P$$ that may be used to perform the row operation $$R_3 \leftrightarrow R_2$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_1 \to 2R_1$$ and then $$R_3 \leftrightarrow R_2$$ to $$A$$ (note the order).

1. $$Q= \left[\begin{array}{cccc} 2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$PQA$$

#### Example 23 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_4 \to 2R_4$$.
2. Give a $$4 \times 4$$ matrix $$N$$ that may be used to perform the row operation $$R_4 \leftrightarrow R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_4 \to 2R_4$$ and then $$R_4 \leftrightarrow R_1$$ to $$A$$ (note the order).

1. $$Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 2 \end{array}\right]$$
2. $$N= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right]$$
3. $$NQA$$

#### Example 24 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$N$$ that may be used to perform the row operation $$R_2 \to R_2 + 3R_3$$.
2. Give a $$4 \times 4$$ matrix $$B$$ that may be used to perform the row operation $$R_2 \leftrightarrow R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_2 \to R_2 + 3R_3$$ and then $$R_2 \leftrightarrow R_1$$ to $$A$$ (note the order).

1. $$N= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 3 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$B= \left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$BNA$$

#### Example 25 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$P$$ that may be used to perform the row operation $$R_3 \to R_3 + 4R_2$$.
2. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_4 \to 4R_4$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_4 \to 4R_4$$ and then $$R_3 \to R_3 + 4R_2$$ to $$A$$ (note the order).

1. $$P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 4 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 4 \end{array}\right]$$
3. $$PQA$$

#### Example 26 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_3 \to -3R_3$$.
2. Give a $$4 \times 4$$ matrix $$N$$ that may be used to perform the row operation $$R_4 \leftrightarrow R_2$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_3 \to -3R_3$$ and then $$R_4 \leftrightarrow R_2$$ to $$A$$ (note the order).

1. $$C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -3 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$N= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \end{array}\right]$$
3. $$NCA$$

#### Example 27 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$M$$ that may be used to perform the row operation $$R_2 \leftrightarrow R_1$$.
2. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_2 \to R_2 + -4R_4$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_2 \to R_2 + -4R_4$$ and then $$R_2 \leftrightarrow R_1$$ to $$A$$ (note the order).

1. $$M= \left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -4 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$MCA$$

#### Example 28 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$N$$ that may be used to perform the row operation $$R_4 \to R_4 + -3R_3$$.
2. Give a $$4 \times 4$$ matrix $$M$$ that may be used to perform the row operation $$R_1 \to 4R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_1 \to 4R_1$$ and then $$R_4 \to R_4 + -3R_3$$ to $$A$$ (note the order).

1. $$N= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & -3 & 1 \end{array}\right]$$
2. $$M= \left[\begin{array}{cccc} 4 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$NMA$$

#### Example 29 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$B$$ that may be used to perform the row operation $$R_1 \to 5R_1$$.
2. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_4 \to R_4 + -5R_2$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_4 \to R_4 + -5R_2$$ and then $$R_1 \to 5R_1$$ to $$A$$ (note the order).

1. $$B= \left[\begin{array}{cccc} 5 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & -5 & 0 & 1 \end{array}\right]$$
3. $$BCA$$

#### Example 30 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_4 \to R_4 + -2R_3$$.
2. Give a $$4 \times 4$$ matrix $$M$$ that may be used to perform the row operation $$R_2 \to 3R_2$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_4 \to R_4 + -2R_3$$ and then $$R_2 \to 3R_2$$ to $$A$$ (note the order).

1. $$Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & -2 & 1 \end{array}\right]$$
2. $$M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$MQA$$

#### Example 31 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_3 \leftrightarrow R_1$$.
2. Give a $$4 \times 4$$ matrix $$M$$ that may be used to perform the row operation $$R_1 \to -3R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_3 \leftrightarrow R_1$$ and then $$R_1 \to -3R_1$$ to $$A$$ (note the order).

1. $$C= \left[\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$M= \left[\begin{array}{cccc} -3 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$MCA$$

#### Example 32 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$P$$ that may be used to perform the row operation $$R_4 \to R_4 + -3R_1$$.
2. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_3 \to 5R_3$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_3 \to 5R_3$$ and then $$R_4 \to R_4 + -3R_1$$ to $$A$$ (note the order).

1. $$P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ -3 & 0 & 0 & 1 \end{array}\right]$$
2. $$C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$PCA$$

#### Example 33 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_3 \leftrightarrow R_1$$.
2. Give a $$4 \times 4$$ matrix $$B$$ that may be used to perform the row operation $$R_4 \to 5R_4$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_4 \to 5R_4$$ and then $$R_3 \leftrightarrow R_1$$ to $$A$$ (note the order).

1. $$Q= \left[\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 5 \end{array}\right]$$
3. $$QBA$$

#### Example 34 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_1 \to 5R_1$$.
2. Give a $$4 \times 4$$ matrix $$P$$ that may be used to perform the row operation $$R_2 \to R_2 + -5R_3$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_1 \to 5R_1$$ and then $$R_2 \to R_2 + -5R_3$$ to $$A$$ (note the order).

1. $$Q= \left[\begin{array}{cccc} 5 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & -5 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$PQA$$

#### Example 35 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$B$$ that may be used to perform the row operation $$R_2 \to R_2 + 5R_3$$.
2. Give a $$4 \times 4$$ matrix $$N$$ that may be used to perform the row operation $$R_4 \leftrightarrow R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_2 \to R_2 + 5R_3$$ and then $$R_4 \leftrightarrow R_1$$ to $$A$$ (note the order).

1. $$B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 5 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$N= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right]$$
3. $$NBA$$

#### Example 36 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_4 \leftrightarrow R_1$$.
2. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_3 \to 4R_3$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_4 \leftrightarrow R_1$$ and then $$R_3 \to 4R_3$$ to $$A$$ (note the order).

1. $$C= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right]$$
2. $$Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 4 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$QCA$$

#### Example 37 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_1 \leftrightarrow R_4$$.
2. Give a $$4 \times 4$$ matrix $$P$$ that may be used to perform the row operation $$R_4 \to -3R_4$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_4 \to -3R_4$$ and then $$R_1 \leftrightarrow R_4$$ to $$A$$ (note the order).

1. $$C= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right]$$
2. $$P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & -3 \end{array}\right]$$
3. $$CPA$$

#### Example 38 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$B$$ that may be used to perform the row operation $$R_1 \to -2R_1$$.
2. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_4 \leftrightarrow R_3$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_1 \to -2R_1$$ and then $$R_4 \leftrightarrow R_3$$ to $$A$$ (note the order).

1. $$B= \left[\begin{array}{cccc} -2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \end{array}\right]$$
3. $$CBA$$

#### Example 39 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_4 \to R_4 + -2R_3$$.
2. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_1 \to 2R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_1 \to 2R_1$$ and then $$R_4 \to R_4 + -2R_3$$ to $$A$$ (note the order).

1. $$C= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & -2 & 1 \end{array}\right]$$
2. $$Q= \left[\begin{array}{cccc} 2 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$CQA$$

#### Example 40 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_1 \to R_1 + 3R_2$$.
2. Give a $$4 \times 4$$ matrix $$P$$ that may be used to perform the row operation $$R_3 \to 5R_3$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_1 \to R_1 + 3R_2$$ and then $$R_3 \to 5R_3$$ to $$A$$ (note the order).

1. $$Q= \left[\begin{array}{cccc} 1 & 3 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 5 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$PQA$$

#### Example 41 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_3 \to R_3 + -4R_4$$.
2. Give a $$4 \times 4$$ matrix $$M$$ that may be used to perform the row operation $$R_2 \leftrightarrow R_3$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_3 \to R_3 + -4R_4$$ and then $$R_2 \leftrightarrow R_3$$ to $$A$$ (note the order).

1. $$Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & -4 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$MQA$$

#### Example 42 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$M$$ that may be used to perform the row operation $$R_2 \leftrightarrow R_3$$.
2. Give a $$4 \times 4$$ matrix $$N$$ that may be used to perform the row operation $$R_2 \to 3R_2$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_2 \to 3R_2$$ and then $$R_2 \leftrightarrow R_3$$ to $$A$$ (note the order).

1. $$M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$N= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 3 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$MNA$$

#### Example 43 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$C$$ that may be used to perform the row operation $$R_3 \leftrightarrow R_1$$.
2. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_4 \to R_4 + -3R_3$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_3 \leftrightarrow R_1$$ and then $$R_4 \to R_4 + -3R_3$$ to $$A$$ (note the order).

1. $$C= \left[\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & -3 & 1 \end{array}\right]$$
3. $$QCA$$

#### Example 44 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$B$$ that may be used to perform the row operation $$R_3 \to -4R_3$$.
2. Give a $$4 \times 4$$ matrix $$M$$ that may be used to perform the row operation $$R_2 \leftrightarrow R_3$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_3 \to -4R_3$$ and then $$R_2 \leftrightarrow R_3$$ to $$A$$ (note the order).

1. $$B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & -4 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$MBA$$

#### Example 45 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$M$$ that may be used to perform the row operation $$R_2 \to R_2 + -2R_4$$.
2. Give a $$4 \times 4$$ matrix $$P$$ that may be used to perform the row operation $$R_3 \leftrightarrow R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_2 \to R_2 + -2R_4$$ and then $$R_3 \leftrightarrow R_1$$ to $$A$$ (note the order).

1. $$M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -2 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$P= \left[\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$PMA$$

#### Example 46 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$B$$ that may be used to perform the row operation $$R_4 \to R_4 + 3R_3$$.
2. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_4 \to 3R_4$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_4 \to 3R_4$$ and then $$R_4 \to R_4 + 3R_3$$ to $$A$$ (note the order).

1. $$B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 3 & 1 \end{array}\right]$$
2. $$Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 3 \end{array}\right]$$
3. $$BQA$$

#### Example 47 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$P$$ that may be used to perform the row operation $$R_2 \to R_2 + -4R_4$$.
2. Give a $$4 \times 4$$ matrix $$B$$ that may be used to perform the row operation $$R_2 \leftrightarrow R_3$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_2 \leftrightarrow R_3$$ and then $$R_2 \to R_2 + -4R_4$$ to $$A$$ (note the order).

1. $$P= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & -4 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$PBA$$

#### Example 48 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$N$$ that may be used to perform the row operation $$R_4 \leftrightarrow R_1$$.
2. Give a $$4 \times 4$$ matrix $$B$$ that may be used to perform the row operation $$R_2 \to R_2 + -2R_3$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_2 \to R_2 + -2R_3$$ and then $$R_4 \leftrightarrow R_1$$ to $$A$$ (note the order).

1. $$N= \left[\begin{array}{cccc} 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 1 & 0 & 0 & 0 \end{array}\right]$$
2. $$B= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & -2 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$NBA$$

#### Example 49 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$Q$$ that may be used to perform the row operation $$R_1 \to R_1 + 4R_4$$.
2. Give a $$4 \times 4$$ matrix $$N$$ that may be used to perform the row operation $$R_2 \leftrightarrow R_1$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_1 \to R_1 + 4R_4$$ and then $$R_2 \leftrightarrow R_1$$ to $$A$$ (note the order).

1. $$Q= \left[\begin{array}{cccc} 1 & 0 & 0 & 4 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$N= \left[\begin{array}{cccc} 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$NQA$$

#### Example 50 π

Let $$A$$ be a $$4 \times 4$$ matrix.

1. Give a $$4 \times 4$$ matrix $$P$$ that may be used to perform the row operation $$R_3 \leftrightarrow R_1$$.
2. Give a $$4 \times 4$$ matrix $$M$$ that may be used to perform the row operation $$R_3 \to 3R_3$$.
3. Use matrix multiplication to describe the matrix obtained by applying $$R_3 \leftrightarrow R_1$$ and then $$R_3 \to 3R_3$$ to $$A$$ (note the order).

1. $$P= \left[\begin{array}{cccc} 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
2. $$M= \left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 3 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$$
3. $$MPA$$