https://file.aiquickdraw.com/mathbot/file/1730497976194tknpaufc.jpg

To find the value of the determinant of the given \( 3 \times 3 \) matrix:

\[
\begin{vmatrix}
-1 & -3 & -2 \\
4 & 7 & 3 \\
3 & 9 & 9 \\
\end{vmatrix}
\]

we can use the formula for the determinant of a \( 3 \times 3 \) matrix:

\[
\text{det} = a(ei - fh) - b(di - fg) + c(dh - eg)
\]

For this matrix:
\[
a = -1, \; b = -3, \; c = -2
\]
\[
d = 4, \; e = 7, \; f = 3
\]
\[
g = 3, \; h = 9, \; i = 9
\]

### Step-by-Step Calculation:

1. **Calculate \( ei - fh \):**
   \[
   ei - fh = (7 \cdot 9) - (3 \cdot 9) = 63 - 27 = 36
   \]

2. **Calculate \( di - fg \):**
   \[
   di - fg = (4 \cdot 9) - (3 \cdot 3) = 36 - 9 = 27
   \]

3. **Calculate \( dh - eg \):**
   \[
   dh - eg = (4 \cdot 9) - (7 \cdot 3) = 36 - 21 = 15
   \]

4. **Combine the results into the determinant formula:**
   \[
   \text{det} = -1(36) - (-3)(27) + (-2)(15)
   \]
   \[
   = -36 + 81 - 30
   \]
   \[
   = 15
   \]

### Final Answer:
The value of the determinant is \( 15 \).

So, the correct answer is:
- **c. 15**

Would you like a deeper explanation or have any questions?

---

Here are 5 related questions for further practice:

1. What is the determinant of a 2x2 matrix, and how is it calculated?
2. How would changing one element in the matrix affect the determinant?
3. How can we calculate the inverse of a 3x3 matrix using its determinant?
4. What properties of determinants make calculations easier for certain types of matrices?
5. How does the determinant relate to the area (in 2D) or volume (in 3D) defined by the matrix?

**Tip:** Remember that a matrix with a zero determinant is singular and does not have an inverse.

Find the value of the determinant of the given 3x3 matrix:

\[
\begin{vmatrix}
-1 & -3 & -2 \\
4 & 7 & 3 \\
3 & 9 & 9 \\
\end{vmatrix}
\]

with the options:
a. 17
b. 12
c. 15
d. 13

Find the value of the determinant for a 3x3 matrix using the determinant formula. This problem is suitable for high school students in Grades 10-12 studying linear algebra and matrix operations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation