Math Problem Statement
answer the question
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
Solution
To find the value of the determinant of the given matrix:
-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.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Matrix Determinants
Linear Algebra
Formulas
Determinant of a 3x3 matrix: det = a(ei - fh) - b(di - fg) + c(dh - eg)
Theorems
Determinant Calculation Formula for 3x3 Matrix
Suitable Grade Level
Grades 10-12