Math Problem Statement
Solution
The image presents two matrices, and you're asked to find their determinants using properties. Let’s address each matrix:
Matrix (a):
1 & 4 & 2 \\ 0 & 0 & 0 \\ 3 & 9 & 5 \end{vmatrix}$$ **Step 1: Analyze the matrix** Since the second row consists entirely of zeros, **the determinant of this matrix is zero**. This is a standard property: if any row or column of a matrix is entirely composed of zeros, the determinant of the matrix is zero. So, for matrix (a): $$\boxed{0}$$ ### Matrix (b): $$\begin{vmatrix} 1 & 4 & 2 \\ 1 & 4 & 2 \\ 3 & 9 & 5 \end{vmatrix}$$ **Step 1: Row duplication** Here, rows 1 and 2 are identical. According to the properties of determinants, if two rows (or columns) of a matrix are identical, the determinant is **zero**. So, for matrix (b): $$\boxed{0}$$ Would you like further details or explanations? ### Follow-up Questions: 1. How do row operations affect the determinant? 2. What happens to the determinant when two rows are swapped? 3. Can you calculate the determinant of a triangular matrix easily? 4. What are the properties of determinants when adding multiples of rows? 5. How does scaling a row by a scalar affect the determinant? ### Tip: Always check for row or column simplifications (zeros or identical rows/columns) to quickly determine if the determinant is zero!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Matrix Properties
Formulas
Determinant of a matrix
Property: Determinant is zero if a row or column is zero
Property: Determinant is zero if two rows are identical
Theorems
If any row (or column) of a matrix is entirely composed of zeros, the determinant is zero.
If two rows (or columns) of a matrix are identical, the determinant is zero.
Suitable Grade Level
Grades 10-12
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