Math Problem Statement
Based on the matrix below, find the following: 𝐴 = | 1 2 −1 | | 2 1 2 | |−1 2 1 |
a) Determinant of the Matrix
Solution
To find the determinant of the matrix , we can use the following formula:
For the given matrix :
1 & 2 & -1 \\ 2 & 1 & 2 \\ -1 & 2 & 1 \end{pmatrix}$$ Here, $$a = 1$$, $$b = 2$$, $$c = -1$$, $$d = 2$$, $$e = 1$$, $$f = 2$$, $$g = -1$$, $$h = 2$$, $$i = 1$$. Substituting these values into the formula: $$\text{det}(A) = 1(1 \cdot 1 - 2 \cdot 2) - 2(2 \cdot 1 - 2 \cdot -1) + (-1)(2 \cdot 2 - 1 \cdot -1)$$ Let's calculate each part: $$\text{det}(A) = 1(1 - 4) - 2(2 + 2) + (-1)(4 + 1)$$ $$\text{det}(A) = 1(-3) - 2(4) + (-1)(5)$$ $$\text{det}(A) = -3 - 8 - 5 = -16$$ So, the determinant of matrix $$A$$ is $$\boxed{-16}$$. Would you like more details or have any other questions? Here are 8 follow-up questions that you might find helpful: 1. How does the determinant of a matrix affect its invertibility? 2. What are the properties of determinants that are useful in matrix operations? 3. How does the determinant relate to the volume of a parallelepiped in geometry? 4. What happens to the determinant if two rows of a matrix are swapped? 5. How can the determinant be used to solve systems of linear equations? 6. What is the significance of a zero determinant? 7. How do you compute the determinant of a larger $$n \times n$$ matrix? 8. Can the determinant be used to find eigenvalues of a matrix? **Tip:** Always remember that the determinant of a matrix provides important insights into the properties of the matrix, including whether it is invertible and the scaling factor of the linear transformation represented by the matrix.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Algebra
Determinants
Formulas
Formula for determinant of a 3x3 matrix
Theorems
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Suitable Grade Level
Grades 10-12