Math Problem Statement
Calculate the value of the determinant of the given 3x3 matrix: [[1, -1, 3], [2, 5, 0], [-3, 1, 2]].
Solution
To find the value of the determinant of the given matrix:
1 & -1 & 3 \\ 2 & 5 & 0 \\ -3 & 1 & 2 \end{vmatrix}$$ We will use the formula for the determinant of a $$3 \times 3$$ matrix, given by: $$\text{Det} = a(ei - fh) - b(di - fg) + c(dh - eg)$$ where: - $$a = 1$$, $$b = -1$$, $$c = 3$$ - $$d = 2$$, $$e = 5$$, $$f = 0$$ - $$g = -3$$, $$h = 1$$, $$i = 2$$ ### Step-by-step Calculation: 1. Compute $$ei - fh$$: $$ei - fh = (5 \times 2) - (0 \times 1) = 10 - 0 = 10$$ 2. Compute $$di - fg$$: $$di - fg = (2 \times 2) - (0 \times -3) = 4 - 0 = 4$$ 3. Compute $$dh - eg$$: $$dh - eg = (2 \times 1) - (5 \times -3) = 2 + 15 = 17$$ 4. Substitute these values into the determinant formula: $$\text{Det} = 1 \times 10 - (-1) \times 4 + 3 \times 17$$ Simplifying further: $$\text{Det} = 10 + 4 + 51 = 65$$ ### Final Answer: $$\text{Determinant} = 65$$ Would you like further explanation or have questions about the steps? Here are five related questions to consider: 1. How do you calculate the determinant of larger matrices, such as $$4 \times 4$$ matrices? 2. What properties of determinants are useful in simplifying matrix calculations? 3. How can determinants be used to determine if a matrix is invertible? 4. What is the geometric interpretation of a determinant? 5. How are determinants applied in solving systems of linear equations? **Tip:** Determinants can be sensitive to row operations; be careful when simplifying matrices as it can affect the determinant's value.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Formulas
Det = a(ei - fh) - b(di - fg) + c(dh - eg)
Theorems
Properties of Determinants
Suitable Grade Level
Grades 10-12
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