How to Calculate the Determinant of a Triangular Matrix

Math Problem Statement

What is the determinant of the matrix A = [[1, 3, -5], [0, 1, 2], [0, 0, 1]]?

The given problem asks to find the determinant of the matrix $A$ :

1 & 3 & -5 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{pmatrix}.$$ ### Step-by-Step Solution: The determinant of a $$3 \times 3$$ matrix can be calculated using the general formula, but here, since the matrix $$A$$ is an upper triangular matrix (all elements below the main diagonal are 0), the determinant is simply the product of the diagonal elements: $$\text{det}(A) = (1) \cdot (1) \cdot (1) = 1.$$ ### Final Answer: $$\text{det}(A) = 1$$ Let me know if you would like further explanation or have additional questions! --- Here are five related questions to deepen your understanding: 1. What is the general formula for the determinant of a $$3 \times 3$$ matrix? 2. How can you determine if a matrix is upper triangular? 3. What properties of determinants simplify the calculation for triangular matrices? 4. If one diagonal element were zero in a triangular matrix, what would happen to the determinant? 5. Can a triangular matrix ever have a negative determinant? **Tip:** Always check the matrix type (e.g., triangular, diagonal) before solving for determinants to save time!

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Linear Algebra
Matrix Determinants
Triangular Matrices

Determinant of a triangular matrix is the product of its diagonal elements

Determinant of triangular matrices

Grades 11-12 or college-level linear algebra

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