Math Problem Statement
What is the determinant of the matrix A = [[1, 3, -5], [0, 1, 2], [0, 0, 1]]?
Solution
To solve for the determinant of matrix :
The given matrix is:
1 & 3 & -5 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{bmatrix}$$ ### Steps to Compute $$\det(A)$$: 1. **Identify Matrix Properties**: This is an upper triangular matrix, as all elements below the main diagonal are zeros. 2. **Determinant of an Upper Triangular Matrix**: The determinant of an upper triangular matrix is the product of the diagonal entries. Thus, $$\det(A) = (1)(1)(1) = 1$$ ### Final Answer: $$\boxed{1}$$ Let me know if you'd like detailed explanations of triangular matrices or determinants! Here are 5 related questions for further learning: 1. What is the determinant of a lower triangular matrix? 2. How do we compute the determinant of a 4x4 matrix? 3. What are the properties of determinants regarding row or column swaps? 4. How does scaling a row affect the determinant of a matrix? 5. What is the determinant of a singular matrix? **Tip**: Always check for matrix structure (triangular, diagonal) to simplify determinant calculations.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Determinants
Triangular Matrices
Formulas
Determinant of an upper triangular matrix is the product of its diagonal entries
Theorems
Properties of Determinants
Suitable Grade Level
Undergraduate (Entry-level) or Advanced High School (Grades 11-12)