Math Problem Statement
Find |A| if A = [[1, 2, 3], [0, 1, 0], [0, 3, 0]]
Solution
The problem asks to find the determinant of the matrix , where:
1 & 2 & 3 \\ 0 & 1 & 0 \\ 0 & 3 & 0 \end{bmatrix}.$$ ### Steps to Solve 1. **Matrix Determinant Formula**: For a $$3 \times 3$$ matrix, the determinant is calculated as: $$|A| = a(ei − fh) − b(di − fg) + c(dh − eg),$$ where $$A = \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix}$$. Here: $$a = 1, b = 2, c = 3, d = 0, e = 1, f = 0, g = 0, h = 3, i = 0.$$ 2. **Expanding Along the First Row**: Using cofactor expansion along the first row: $$|A| = 1 \cdot \begin{vmatrix} 1 & 0 \\ 3 & 0 \end{vmatrix} - 2 \cdot \begin{vmatrix} 0 & 0 \\ 0 & 0 \end{vmatrix} + 3 \cdot \begin{vmatrix} 0 & 1 \\ 0 & 3 \end{vmatrix}.$$ 3. **Evaluate the 2x2 Determinants**: - For $$\begin{vmatrix} 1 & 0 \\ 3 & 0 \end{vmatrix}$$: $$(1 \cdot 0) - (0 \cdot 3) = 0.$$ - For $$\begin{vmatrix} 0 & 0 \\ 0 & 0 \end{vmatrix}$$: $$(0 \cdot 0) - (0 \cdot 0) = 0.$$ - For $$\begin{vmatrix} 0 & 1 \\ 0 & 3 \end{vmatrix}$$: $$(0 \cdot 3) - (1 \cdot 0) = 0.$$ 4. **Substitute Back**: $$|A| = 1(0) - 2(0) + 3(0) = 0.$$ ### Final Answer: $$|A| = 0$$ Would you like me to explain a specific step in more detail? --- ### Related Questions 1. What is the significance of a zero determinant for a matrix? 2. How do you compute determinants for $$n \times n$$ matrices (where $$n > 3$$)? 3. What is the cofactor expansion method for finding determinants? 4. How do row and column operations affect the determinant of a matrix? 5. What is the determinant’s role in determining matrix invertibility? ### Tip: If any row or column of a matrix is entirely zeros, the determinant of the matrix will always be zero.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Determinants
Cofactor Expansion
Formulas
Determinant formula for 3x3 matrices
Theorems
Cofactor Expansion Theorem
Suitable Grade Level
Grades 11-12