M2 - Row operations as matrix multiplication

Example 1 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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 🔗

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. $$MPA$$