Determinants of 3x3 Matrices worksheet with 12 problems to evaluate determinants, featuring matrices with variables and numbers.

To solve the problem of evaluating the determinants of the given 3x3 matrices, we will use the formula for the determinant of a 3x3 matrix:

For a matrix \( A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \), the determinant is calculated as:
\[
\det(A) = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

Let's evaluate each matrix step by step.

---

1. Matrix:

\[
\begin{bmatrix}
-6 & -5 & 0 \\
-4 & -3 & 4 \\
-5 & -6 & 6
\end{bmatrix}
\]

Using the determinant formula:
\[
\det(A) = (-6)((-3)(6) - (4)(-6)) - (-5)((-4)(6) - (4)(-5)) + 0((-4)(-6) - (-3)(-5))
\]
\[
= (-6)((-18) - (-24)) - (-5)((-24) - (-20)) + 0((24) - (15))
\]
\[
= (-6)(-18 + 24) - (-5)(-24 + 20) + 0(24 - 15)
\]
\[
= (-6)(6) - (-5)(-4) + 0(9)
\]
\[
= -36 - 20 + 0
\]
\[
= -56
\]

Answer:
\[
\boxed{-56}
\]

---

2. Matrix:

\[
\begin{bmatrix}
3 & s+5 & 2 \\
-3 & -2 & 4 \\
s-1 & 4 & 3
\end{bmatrix}, \quad s = 3
\]

Substitute \( s = 3 \):
\[
\begin{bmatrix}
3 & 8 & 2 \\
-3 & -2 & 4 \\
2 & 4 & 3
\end{bmatrix}
\]

Using the determinant formula:
\[
\det(A) = 3((-2)(3) - (4)(4)) - 8((-3)(3) - (4)(2)) + 2((-3)(4) - (-2)(2))
\]
\[
= 3((-6) - 16) - 8((-9) - 8) + 2((-12) - (-4))
\]
\[
= 3(-22) - 8(-17) + 2(-12 + 4)
\]
\[
= 3(-22) - 8(-17) + 2(-8)
\]
\[
= -66 + 136 - 16
\]
\[
= 54
\]

Answer:
\[
\boxed{54}
\]

---

3. Matrix:

\[
\begin{bmatrix}
-5 & -1 & 4 \\
-2 & 2 & -3 \\
3 & 4 & 6
\end{bmatrix}
\]

Using the determinant formula:
\[
\det(A) = (-5)((2)(6) - (-3)(4)) - (-1)((-2)(6) - (-3)(3)) + 4((-2)(4) - (2)(3))
\]
\[
= (-5)((12) - (-12)) - (-1)((-12) - (-9)) + 4((-8) - 6)
\]
\[
= (-5)(12 + 12) - (-1)(-12 + 9) + 4(-8 - 6)
\]
\[
= (-5)(24) - (-1)(-3) + 4(-14)
\]
\[
= -120 - 3 - 56
\]
\[
= -179
\]

Answer:
\[
\boxed{-179}
\]

---

4. Matrix:

\[
\begin{bmatrix}
-2 & -4 & 2 \\
3n & -3 & -3n+2 \\
-3 & 2 & -4
\end{bmatrix}, \quad n = 3
\]

Substitute \( n = 3 \):
\[
\begin{bmatrix}
-2 & -4 & 2 \\
9 & -3 & -7 \\
-3 & 2 & -4
\end{bmatrix}
\]

Using the determinant formula:
\[
\det(A) = (-2)((-3)(-4) - (-7)(2)) - (-4)((9)(-4) - (-7)(-3)) + 2((9)(2) - (-3)(-3))
\]
\[
= (-2)((12) - (-14)) - (-4)((-36) - (21)) + 2((18) - (9))
\]
\[
= (-2)(12 + 14) - (-4)(-36 - 21) + 2(18 - 9)
\]
\[
= (-2)(26) - (-4)(-57) + 2(9)
\]
\[
= -52 - 228 + 18
\]
\[
= -262
\]

Answer:
\[
\boxed{-262}
\]

---

5. Matrix:

\[
\begin{bmatrix}
-2k & 2 & -4 \\
2 & -4 & 3k \\
-3 & 4 & 4
\end{bmatrix}, \quad k = 4
\]

Substitute \( k = 4 \):
\[
\begin{bmatrix}
-8 & 2 & -4 \\
2 & -4 & 12 \\
-3 & 4 & 4
\end{bmatrix}
\]

Using the determinant formula:
\[
\det(A) = (-8)((-4)(4) - (12)(4)) - 2((2)(4) - (12)(-3)) + (-4)((2)(4) - (-4)(-3))
\]
\[
= (-8)((-16) - 48) - 2((8) - (-36)) + (-4)((8) - 12)
\]
\[
= (-8)(-16 - 48) - 2(8 + 36) + (-4)(8 - 12)
\]
\[
= (-8)(-64) - 2(44) + (-4)(-4)
\]
\[
= 512 - 88 + 16
\]
\[
= 440
\]

Answer:
\[
\boxed{440}
\]

---

6. Matrix:

\[
\begin{bmatrix}
4 & -1 & 2 \\
-3 & 1 & -2 \\
6 & -1 & -3
\end{bmatrix}
\]

Using the determinant formula:
\[
\det(A) = 4((1)(-3) - (-2)(-1)) - (-1)((-3)(-3) - (-2)(6)) + 2((-3)(-1) - (1)(6))
\]
\[
= 4((-3) - 2) - (-1)((9) - (-12)) + 2((3) - 6)
\]
\[
= 4(-3 - 2) - (-1)(9 + 12) + 2(3 - 6)
\]
\[
= 4(-5) - (-1)(21) + 2(-3)
\]
\[
= -20 + 21 - 6
\]
\[
= -5
\]

Answer:
\[
\boxed{-5}
\]

---

7. Matrix:

\[
\begin{bmatrix}
2 & -4 & -4 \\
-3z & 3 & -4z+4 \\
4 & 4 & -3
\end{bmatrix}, \quad z = 3
\]

Substitute \( z = 3 \):
\[
\begin{bmatrix}
2 & -4 & -4 \\
-9 & 3 & -8 \\
4 & 4 & -3
\end{bmatrix}
\]

Using the determinant formula:
\[
\det(A) = 2((3)(-3) - (-8)(4)) - (-4)((-9)(-3) - (-8)(4)) + (-4)((-9)(4) - (3)(4))
\]
\[
= 2((-9) - (-32)) - (-4)((27) - (-32)) + (-4)((-36) - 12)
\]
\[
= 2(-9 + 32) - (-4)(27 + 32) + (-4)(-36 - 12)
\]
\[
= 2(23) - (-4)(59) + (-4)(-48)
\]
\[
= 46 + 236 + 192
\]
\[
= 474
\]

Answer:
\[
\boxed{474}
\]

---

8. Matrix:

\[
\begin{bmatrix}
1 & -5 & 6 \\
4 & -5 & -4 \\
0 & 5 & -6
\end{bmatrix}
\]

Using the determinant formula:
\[
\det(A) = 1((-5)(-6) - (-4)(5)) - (-5)((4)(-6) - (-4)(0)) + 6((4)(5) - (-5)(0))
\]
\[
= 1((30) - (-20)) - (-5)((-24) - 0) + 6((20) - 0)
\]
\[
= 1(30 + 20) - (-5)(-24) + 6(20)
\]
\[
= 1(50) - (-5)(-24) + 6(20)
\]
\[
= 50 - 120 + 120
\]
\[
= 50
\]

Answer:
\[
\boxed{50}
\]

---

9. Matrix:

\[
\begin{bmatrix}
-1 & 3 & 4 \\
-6 & -6 & -1 \\
-2 & 2 & -2
\end{bmatrix}
\]

Using the determinant formula:
\[
\det(A) = (-1)((-6)(-2) - (-1)(2)) - 3((-6)(-2) - (-1)(-2)) + 4((-6)(2) - (-6)(-2))
\]
\[
= (-1)((12) - (-2)) - 3((12) - 2) + 4((-12) - 12)
\]
\[
= (-1)(12 + 2) - 3(12 - 2) + 4(-12 - 12)
\]
\[
= (-1)(14) - 3(10) + 4(-24)
\]
\[
= -14 - 30 - 96
\]
\[
= -140
\]

Answer:
\[
\boxed{-140}
\]

---

10. Matrix:

\[
\begin{bmatrix}
6 & -6 & 4 \\
3 & 2 & -2 \\
3 & 5 & -1
\end{bmatrix}
\]

Using the determinant formula:
\[
\det(A) = 6((2)(-1) - (-2)(5)) - (-6)((3)(-1) - (-2)(3)) + 4((3)(5) - (2)(3))
\]
\[
= 6((-2) - (-10)) - (-6)((-3) - (-6)) + 4((15) - 6)
\]
\[
= 6(-2 + 10) - (-6)(-3 + 6) + 4(15 - 6)
\]
\[
= 6(8) - (-6)(3) + 4(9)
\]
\[
= 48 + 18 + 36
\]
\[
= 102
\]

Answer:
\[
\boxed{102}
\]

---

11. Matrix:

\[
\begin{bmatrix}
-3 & 4 & -2 \\
2g & 3 & 4g+2 \\
4 & 3 & -4
\end{bmatrix}, \quad g = -2
\]

Substitute \( g = -2 \):
\[
\begin{bmatrix}
-3 & 4 & -2 \\
-4 & 3 & -6 \\
4 & 3 & -4
\end{bmatrix}
\]

Using the determinant formula:
\[
\det(A) = (-3)((3)(-4) - (-6)(3)) - 4((-4)(-4) - (-6)(4)) + (-2)((-4)(3) - (3)(4))
\]
\[
= (-3)((-12) - (-18)) - 4((16) - (-24)) + (-2)((-12) - 12)
\]
\[
= (-3)(-12 + 18) - 4(16 + 24) + (-2)(-12 - 12)
\]
\[
= (-3)(6) - 4(40) + (-2)(-24)
\]
\[
= -18 - 160 + 48
\]
\[
= -130
\]

Answer:
\[
\boxed{-130}
\]

---

12. Matrix:

\[
\begin{bmatrix}
4b & -4 & -2 \\
-2 & 3 & 4 \\
3+b & 2 & -3
\end{bmatrix}, \quad b = 2
\]

Substitute \( b = 2 \):
\[
\begin{bmatrix}
8 & -4 & -2 \\
-2 & 3 & 4 \\
5 & 2 & -3
\end{bmatrix}
\]

Using the determinant formula:
\[
\det(A) = 8((3)(-3) - (4)(2)) - (-4)((-2)(-3) - (4)(5)) + (-2)((-2)(2) - (3)(5))
\]
\[
= 8((-9) - 8) - (-4)((6) - 20) + (-2)((-4) - 15)
\]
\[
= 8(-9 - 8) - (-4)(6 - 20) + (-2)(-4 - 15)
\]
\[
= 8(-17) - (-4)(-14) + (-2)(-19)
\]
\[
= -136 - 56 + 38
\]
\[
= -154
\]

Answer:
\[
\boxed{-154}
\]

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Final Answers:

\[
\boxed{-56, 54, -179, -262, 440, -5, 474, 50, -140, 102, -130, -154}
\]