## G1 - Row operations and determinants

#### Example 1 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-6$$.

1. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to 4R_1$$. What is $$\operatorname{det}\ P$$?
2. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \leftrightarrow R_4$$. What is $$\operatorname{det}\ Q$$?
3. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to R_4 + 3R_1$$. What is $$\operatorname{det}\ M$$?

1. $$\operatorname{det}\ P= -24$$
2. $$\operatorname{det}\ Q= 6$$
3. $$\operatorname{det}\ M= -6$$

#### Example 2 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$6$$.

1. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to 5R_2$$. What is $$\operatorname{det}\ B$$?
2. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to R_3 + -2R_2$$. What is $$\operatorname{det}\ N$$?
3. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \leftrightarrow R_3$$. What is $$\operatorname{det}\ C$$?

1. $$\operatorname{det}\ B= 30$$
2. $$\operatorname{det}\ N= 6$$
3. $$\operatorname{det}\ C= -6$$

#### Example 3 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-5$$.

1. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to R_3 + 4R_1$$. What is $$\operatorname{det}\ P$$?
2. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \leftrightarrow R_2$$. What is $$\operatorname{det}\ N$$?
3. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to 3R_3$$. What is $$\operatorname{det}\ Q$$?

1. $$\operatorname{det}\ P= -5$$
2. $$\operatorname{det}\ N= 5$$
3. $$\operatorname{det}\ Q= -15$$

#### Example 4 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$3$$.

1. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to 4R_1$$. What is $$\operatorname{det}\ Q$$?
2. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \leftrightarrow R_1$$. What is $$\operatorname{det}\ B$$?
3. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to R_1 + 4R_3$$. What is $$\operatorname{det}\ M$$?

1. $$\operatorname{det}\ Q= 12$$
2. $$\operatorname{det}\ B= -3$$
3. $$\operatorname{det}\ M= 3$$

#### Example 5 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-2$$.

1. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to -5R_4$$. What is $$\operatorname{det}\ B$$?
2. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to R_2 + 2R_4$$. What is $$\operatorname{det}\ Q$$?
3. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \leftrightarrow R_1$$. What is $$\operatorname{det}\ M$$?

1. $$\operatorname{det}\ B= 10$$
2. $$\operatorname{det}\ Q= -2$$
3. $$\operatorname{det}\ M= 2$$

#### Example 6 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-4$$.

1. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to R_3 + -2R_4$$. What is $$\operatorname{det}\ B$$?
2. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to 2R_2$$. What is $$\operatorname{det}\ C$$?
3. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \leftrightarrow R_3$$. What is $$\operatorname{det}\ M$$?

1. $$\operatorname{det}\ B= -4$$
2. $$\operatorname{det}\ C= -8$$
3. $$\operatorname{det}\ M= 4$$

#### Example 7 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-5$$.

1. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \leftrightarrow R_2$$. What is $$\operatorname{det}\ Q$$?
2. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to 4R_3$$. What is $$\operatorname{det}\ C$$?
3. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to R_1 + -3R_2$$. What is $$\operatorname{det}\ B$$?

1. $$\operatorname{det}\ Q= 5$$
2. $$\operatorname{det}\ C= -20$$
3. $$\operatorname{det}\ B= -5$$

#### Example 8 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-4$$.

1. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to R_2 + -5R_3$$. What is $$\operatorname{det}\ M$$?
2. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \leftrightarrow R_2$$. What is $$\operatorname{det}\ P$$?
3. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to -2R_3$$. What is $$\operatorname{det}\ B$$?

1. $$\operatorname{det}\ M= -4$$
2. $$\operatorname{det}\ P= 4$$
3. $$\operatorname{det}\ B= 8$$

#### Example 9 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-7$$.

1. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to R_4 + -5R_1$$. What is $$\operatorname{det}\ N$$?
2. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to 5R_4$$. What is $$\operatorname{det}\ B$$?
3. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \leftrightarrow R_3$$. What is $$\operatorname{det}\ P$$?

1. $$\operatorname{det}\ N= -7$$
2. $$\operatorname{det}\ B= -35$$
3. $$\operatorname{det}\ P= 7$$

#### Example 10 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$2$$.

1. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \leftrightarrow R_2$$. What is $$\operatorname{det}\ C$$?
2. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to -2R_3$$. What is $$\operatorname{det}\ M$$?
3. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to R_1 + 5R_4$$. What is $$\operatorname{det}\ N$$?

1. $$\operatorname{det}\ C= -2$$
2. $$\operatorname{det}\ M= -4$$
3. $$\operatorname{det}\ N= 2$$

#### Example 11 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-2$$.

1. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \leftrightarrow R_3$$. What is $$\operatorname{det}\ P$$?
2. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to R_1 + 2R_3$$. What is $$\operatorname{det}\ M$$?
3. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to -3R_1$$. What is $$\operatorname{det}\ B$$?

1. $$\operatorname{det}\ P= 2$$
2. $$\operatorname{det}\ M= -2$$
3. $$\operatorname{det}\ B= 6$$

#### Example 12 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$5$$.

1. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to R_4 + -3R_3$$. What is $$\operatorname{det}\ B$$?
2. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \leftrightarrow R_3$$. What is $$\operatorname{det}\ N$$?
3. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to 4R_2$$. What is $$\operatorname{det}\ P$$?

1. $$\operatorname{det}\ B= 5$$
2. $$\operatorname{det}\ N= -5$$
3. $$\operatorname{det}\ P= 20$$

#### Example 13 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$7$$.

1. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to R_3 + 2R_4$$. What is $$\operatorname{det}\ N$$?
2. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \leftrightarrow R_3$$. What is $$\operatorname{det}\ Q$$?
3. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to -2R_1$$. What is $$\operatorname{det}\ M$$?

1. $$\operatorname{det}\ N= 7$$
2. $$\operatorname{det}\ Q= -7$$
3. $$\operatorname{det}\ M= -14$$

#### Example 14 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-5$$.

1. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to R_3 + -4R_4$$. What is $$\operatorname{det}\ C$$?
2. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \leftrightarrow R_1$$. What is $$\operatorname{det}\ B$$?
3. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to -5R_2$$. What is $$\operatorname{det}\ N$$?

1. $$\operatorname{det}\ C= -5$$
2. $$\operatorname{det}\ B= 5$$
3. $$\operatorname{det}\ N= 25$$

#### Example 15 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$6$$.

1. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \leftrightarrow R_4$$. What is $$\operatorname{det}\ C$$?
2. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to R_2 + -4R_4$$. What is $$\operatorname{det}\ N$$?
3. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to 2R_3$$. What is $$\operatorname{det}\ Q$$?

1. $$\operatorname{det}\ C= -6$$
2. $$\operatorname{det}\ N= 6$$
3. $$\operatorname{det}\ Q= 12$$

#### Example 16 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-6$$.

1. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \leftrightarrow R_1$$. What is $$\operatorname{det}\ N$$?
2. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to R_4 + -3R_3$$. What is $$\operatorname{det}\ P$$?
3. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to 5R_4$$. What is $$\operatorname{det}\ M$$?

1. $$\operatorname{det}\ N= 6$$
2. $$\operatorname{det}\ P= -6$$
3. $$\operatorname{det}\ M= -30$$

#### Example 17 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$2$$.

1. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \leftrightarrow R_3$$. What is $$\operatorname{det}\ Q$$?
2. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to R_3 + -5R_4$$. What is $$\operatorname{det}\ M$$?
3. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to -2R_1$$. What is $$\operatorname{det}\ P$$?

1. $$\operatorname{det}\ Q= -2$$
2. $$\operatorname{det}\ M= 2$$
3. $$\operatorname{det}\ P= -4$$

#### Example 18 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-6$$.

1. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \leftrightarrow R_1$$. What is $$\operatorname{det}\ Q$$?
2. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to R_1 + 4R_4$$. What is $$\operatorname{det}\ N$$?
3. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to 2R_4$$. What is $$\operatorname{det}\ C$$?

1. $$\operatorname{det}\ Q= 6$$
2. $$\operatorname{det}\ N= -6$$
3. $$\operatorname{det}\ C= -12$$

#### Example 19 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$3$$.

1. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to R_3 + -2R_2$$. What is $$\operatorname{det}\ P$$?
2. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to -5R_1$$. What is $$\operatorname{det}\ C$$?
3. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \leftrightarrow R_3$$. What is $$\operatorname{det}\ M$$?

1. $$\operatorname{det}\ P= 3$$
2. $$\operatorname{det}\ C= -15$$
3. $$\operatorname{det}\ M= -3$$

#### Example 20 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-5$$.

1. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \leftrightarrow R_3$$. What is $$\operatorname{det}\ Q$$?
2. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to R_2 + -3R_4$$. What is $$\operatorname{det}\ B$$?
3. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to -2R_3$$. What is $$\operatorname{det}\ N$$?

1. $$\operatorname{det}\ Q= 5$$
2. $$\operatorname{det}\ B= -5$$
3. $$\operatorname{det}\ N= 10$$

#### Example 21 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$3$$.

1. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to R_1 + 2R_4$$. What is $$\operatorname{det}\ P$$?
2. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to 2R_2$$. What is $$\operatorname{det}\ M$$?
3. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \leftrightarrow R_2$$. What is $$\operatorname{det}\ Q$$?

1. $$\operatorname{det}\ P= 3$$
2. $$\operatorname{det}\ M= 6$$
3. $$\operatorname{det}\ Q= -3$$

#### Example 22 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-4$$.

1. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to R_4 + -3R_3$$. What is $$\operatorname{det}\ N$$?
2. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to -2R_1$$. What is $$\operatorname{det}\ M$$?
3. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \leftrightarrow R_2$$. What is $$\operatorname{det}\ B$$?

1. $$\operatorname{det}\ N= -4$$
2. $$\operatorname{det}\ M= 8$$
3. $$\operatorname{det}\ B= 4$$

#### Example 23 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$5$$.

1. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to 4R_1$$. What is $$\operatorname{det}\ P$$?
2. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to R_4 + -3R_2$$. What is $$\operatorname{det}\ B$$?
3. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \leftrightarrow R_4$$. What is $$\operatorname{det}\ N$$?

1. $$\operatorname{det}\ P= 20$$
2. $$\operatorname{det}\ B= 5$$
3. $$\operatorname{det}\ N= -5$$

#### Example 24 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-5$$.

1. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \leftrightarrow R_3$$. What is $$\operatorname{det}\ N$$?
2. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to R_2 + 3R_3$$. What is $$\operatorname{det}\ Q$$?
3. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to 4R_3$$. What is $$\operatorname{det}\ C$$?

1. $$\operatorname{det}\ N= 5$$
2. $$\operatorname{det}\ Q= -5$$
3. $$\operatorname{det}\ C= -20$$

#### Example 25 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-4$$.

1. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \leftrightarrow R_4$$. What is $$\operatorname{det}\ C$$?
2. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to -3R_1$$. What is $$\operatorname{det}\ M$$?
3. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to R_3 + -4R_4$$. What is $$\operatorname{det}\ P$$?

1. $$\operatorname{det}\ C= 4$$
2. $$\operatorname{det}\ M= 12$$
3. $$\operatorname{det}\ P= -4$$

#### Example 26 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$2$$.

1. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \leftrightarrow R_4$$. What is $$\operatorname{det}\ C$$?
2. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to R_4 + 5R_3$$. What is $$\operatorname{det}\ Q$$?
3. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to -5R_1$$. What is $$\operatorname{det}\ P$$?

1. $$\operatorname{det}\ C= -2$$
2. $$\operatorname{det}\ Q= 2$$
3. $$\operatorname{det}\ P= -10$$

#### Example 27 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-6$$.

1. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to 5R_2$$. What is $$\operatorname{det}\ Q$$?
2. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \leftrightarrow R_3$$. What is $$\operatorname{det}\ N$$?
3. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to R_2 + 2R_4$$. What is $$\operatorname{det}\ C$$?

1. $$\operatorname{det}\ Q= -30$$
2. $$\operatorname{det}\ N= 6$$
3. $$\operatorname{det}\ C= -6$$

#### Example 28 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-6$$.

1. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to 4R_3$$. What is $$\operatorname{det}\ N$$?
2. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \leftrightarrow R_3$$. What is $$\operatorname{det}\ P$$?
3. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to R_3 + -2R_2$$. What is $$\operatorname{det}\ C$$?

1. $$\operatorname{det}\ N= -24$$
2. $$\operatorname{det}\ P= 6$$
3. $$\operatorname{det}\ C= -6$$

#### Example 29 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-6$$.

1. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \leftrightarrow R_2$$. What is $$\operatorname{det}\ C$$?
2. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to R_2 + 4R_4$$. What is $$\operatorname{det}\ M$$?
3. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to -5R_3$$. What is $$\operatorname{det}\ P$$?

1. $$\operatorname{det}\ C= 6$$
2. $$\operatorname{det}\ M= -6$$
3. $$\operatorname{det}\ P= 30$$

#### Example 30 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$5$$.

1. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \leftrightarrow R_1$$. What is $$\operatorname{det}\ B$$?
2. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to R_1 + -5R_4$$. What is $$\operatorname{det}\ P$$?
3. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to -3R_1$$. What is $$\operatorname{det}\ Q$$?

1. $$\operatorname{det}\ B= -5$$
2. $$\operatorname{det}\ P= 5$$
3. $$\operatorname{det}\ Q= -15$$

#### Example 31 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$3$$.

1. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \leftrightarrow R_4$$. What is $$\operatorname{det}\ B$$?
2. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to R_2 + 3R_3$$. What is $$\operatorname{det}\ N$$?
3. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to -5R_3$$. What is $$\operatorname{det}\ Q$$?

1. $$\operatorname{det}\ B= -3$$
2. $$\operatorname{det}\ N= 3$$
3. $$\operatorname{det}\ Q= -15$$

#### Example 32 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-2$$.

1. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \leftrightarrow R_3$$. What is $$\operatorname{det}\ Q$$?
2. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to R_1 + -2R_3$$. What is $$\operatorname{det}\ B$$?
3. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to 4R_3$$. What is $$\operatorname{det}\ N$$?

1. $$\operatorname{det}\ Q= 2$$
2. $$\operatorname{det}\ B= -2$$
3. $$\operatorname{det}\ N= -8$$

#### Example 33 🔗

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

1. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to R_1 + 4R_3$$. What is $$\operatorname{det}\ P$$?
2. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to 5R_2$$. What is $$\operatorname{det}\ B$$?
3. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \leftrightarrow R_4$$. What is $$\operatorname{det}\ C$$?

1. $$\operatorname{det}\ P= 4$$
2. $$\operatorname{det}\ B= 20$$
3. $$\operatorname{det}\ C= -4$$

#### Example 34 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-6$$.

1. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to -3R_2$$. What is $$\operatorname{det}\ Q$$?
2. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to R_1 + 4R_4$$. What is $$\operatorname{det}\ M$$?
3. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \leftrightarrow R_2$$. What is $$\operatorname{det}\ B$$?

1. $$\operatorname{det}\ Q= 18$$
2. $$\operatorname{det}\ M= -6$$
3. $$\operatorname{det}\ B= 6$$

#### Example 35 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-5$$.

1. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to -2R_2$$. What is $$\operatorname{det}\ B$$?
2. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \leftrightarrow R_3$$. What is $$\operatorname{det}\ N$$?
3. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to R_3 + -3R_4$$. What is $$\operatorname{det}\ M$$?

1. $$\operatorname{det}\ B= 10$$
2. $$\operatorname{det}\ N= 5$$
3. $$\operatorname{det}\ M= -5$$

#### Example 36 🔗

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

1. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to 3R_1$$. What is $$\operatorname{det}\ Q$$?
2. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \leftrightarrow R_1$$. What is $$\operatorname{det}\ C$$?
3. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to R_1 + -2R_4$$. What is $$\operatorname{det}\ B$$?

1. $$\operatorname{det}\ Q= 12$$
2. $$\operatorname{det}\ C= -4$$
3. $$\operatorname{det}\ B= 4$$

#### Example 37 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-4$$.

1. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \leftrightarrow R_1$$. What is $$\operatorname{det}\ Q$$?
2. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to 3R_2$$. What is $$\operatorname{det}\ C$$?
3. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to R_1 + -3R_3$$. What is $$\operatorname{det}\ N$$?

1. $$\operatorname{det}\ Q= 4$$
2. $$\operatorname{det}\ C= -12$$
3. $$\operatorname{det}\ N= -4$$

#### Example 38 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-2$$.

1. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to R_4 + 4R_2$$. What is $$\operatorname{det}\ B$$?
2. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \leftrightarrow R_4$$. What is $$\operatorname{det}\ Q$$?
3. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to -4R_3$$. What is $$\operatorname{det}\ C$$?

1. $$\operatorname{det}\ B= -2$$
2. $$\operatorname{det}\ Q= 2$$
3. $$\operatorname{det}\ C= 8$$

#### Example 39 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-5$$.

1. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to 5R_2$$. What is $$\operatorname{det}\ Q$$?
2. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to R_2 + -4R_4$$. What is $$\operatorname{det}\ M$$?
3. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \leftrightarrow R_2$$. What is $$\operatorname{det}\ P$$?

1. $$\operatorname{det}\ Q= -25$$
2. $$\operatorname{det}\ M= -5$$
3. $$\operatorname{det}\ P= 5$$

#### Example 40 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-4$$.

1. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to R_2 + 2R_4$$. What is $$\operatorname{det}\ Q$$?
2. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \leftrightarrow R_1$$. What is $$\operatorname{det}\ C$$?
3. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to 3R_2$$. What is $$\operatorname{det}\ B$$?

1. $$\operatorname{det}\ Q= -4$$
2. $$\operatorname{det}\ C= 4$$
3. $$\operatorname{det}\ B= -12$$

#### Example 41 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-6$$.

1. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \leftrightarrow R_1$$. What is $$\operatorname{det}\ N$$?
2. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to 4R_2$$. What is $$\operatorname{det}\ Q$$?
3. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to R_2 + -4R_3$$. What is $$\operatorname{det}\ B$$?

1. $$\operatorname{det}\ N= 6$$
2. $$\operatorname{det}\ Q= -24$$
3. $$\operatorname{det}\ B= -6$$

#### Example 42 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-7$$.

1. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to -5R_2$$. What is $$\operatorname{det}\ B$$?
2. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \leftrightarrow R_1$$. What is $$\operatorname{det}\ Q$$?
3. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to R_3 + -5R_2$$. What is $$\operatorname{det}\ M$$?

1. $$\operatorname{det}\ B= 35$$
2. $$\operatorname{det}\ Q= 7$$
3. $$\operatorname{det}\ M= -7$$

#### Example 43 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-5$$.

1. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to R_2 + -4R_3$$. What is $$\operatorname{det}\ B$$?
2. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \leftrightarrow R_2$$. What is $$\operatorname{det}\ Q$$?
3. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to 3R_1$$. What is $$\operatorname{det}\ M$$?

1. $$\operatorname{det}\ B= -5$$
2. $$\operatorname{det}\ Q= 5$$
3. $$\operatorname{det}\ M= -15$$

#### Example 44 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-6$$.

1. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to 4R_1$$. What is $$\operatorname{det}\ C$$?
2. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \leftrightarrow R_2$$. What is $$\operatorname{det}\ M$$?
3. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to R_3 + 3R_4$$. What is $$\operatorname{det}\ B$$?

1. $$\operatorname{det}\ C= -24$$
2. $$\operatorname{det}\ M= 6$$
3. $$\operatorname{det}\ B= -6$$

#### Example 45 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-2$$.

1. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to -5R_2$$. What is $$\operatorname{det}\ N$$?
2. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to R_2 + -5R_4$$. What is $$\operatorname{det}\ Q$$?
3. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \leftrightarrow R_2$$. What is $$\operatorname{det}\ M$$?

1. $$\operatorname{det}\ N= 10$$
2. $$\operatorname{det}\ Q= -2$$
3. $$\operatorname{det}\ M= 2$$

#### Example 46 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$2$$.

1. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to 5R_1$$. What is $$\operatorname{det}\ N$$?
2. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to R_4 + -5R_2$$. What is $$\operatorname{det}\ M$$?
3. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \leftrightarrow R_3$$. What is $$\operatorname{det}\ C$$?

1. $$\operatorname{det}\ N= 10$$
2. $$\operatorname{det}\ M= 2$$
3. $$\operatorname{det}\ C= -2$$

#### Example 47 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$-4$$.

1. Let $$B$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to R_4 + -4R_3$$. What is $$\operatorname{det}\ B$$?
2. Let $$Q$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to 5R_3$$. What is $$\operatorname{det}\ Q$$?
3. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \leftrightarrow R_4$$. What is $$\operatorname{det}\ C$$?

1. $$\operatorname{det}\ B= -4$$
2. $$\operatorname{det}\ Q= -20$$
3. $$\operatorname{det}\ C= 4$$

#### Example 48 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$3$$.

1. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \to R_2 + -2R_1$$. What is $$\operatorname{det}\ M$$?
2. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to -2R_4$$. What is $$\operatorname{det}\ P$$?
3. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \leftrightarrow R_3$$. What is $$\operatorname{det}\ N$$?

1. $$\operatorname{det}\ M= 3$$
2. $$\operatorname{det}\ P= -6$$
3. $$\operatorname{det}\ N= -3$$

#### Example 49 🔗

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

1. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \leftrightarrow R_1$$. What is $$\operatorname{det}\ N$$?
2. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to R_4 + -5R_1$$. What is $$\operatorname{det}\ M$$?
3. Let $$C$$ be the matrix obtained from $$A$$ by applying the row operation $$R_1 \to 3R_1$$. What is $$\operatorname{det}\ C$$?

1. $$\operatorname{det}\ N= -4$$
2. $$\operatorname{det}\ M= 4$$
3. $$\operatorname{det}\ C= 12$$

#### Example 50 🔗

Let $$A$$ be a $$4 \times 4$$ matrix with determinant $$7$$.

1. Let $$M$$ be the matrix obtained from $$A$$ by applying the row operation $$R_4 \to R_4 + 4R_3$$. What is $$\operatorname{det}\ M$$?
2. Let $$P$$ be the matrix obtained from $$A$$ by applying the row operation $$R_3 \to 5R_3$$. What is $$\operatorname{det}\ P$$?
3. Let $$N$$ be the matrix obtained from $$A$$ by applying the row operation $$R_2 \leftrightarrow R_3$$. What is $$\operatorname{det}\ N$$?

1. $$\operatorname{det}\ M= 7$$
2. $$\operatorname{det}\ P= 35$$
3. $$\operatorname{det}\ N= -7$$