Math Problem Statement
Hand-In-Assignment-9.pdf
108.70 KB
problem 3, part 1, do each step by step and then provide the final answer
Solution
Step-by-Step Solution for Problem 3, Part 1
We are tasked to compute the determinant of the matrix:
\begin{bmatrix} 2 & 3 & 4 \\ 5 & 6 & 7 \\ 8 & 9 & 1 \end{bmatrix}.$$ The determinant of a $$3 \times 3$$ matrix can be computed using the cofactor expansion along any row or column. Here, we expand along the **first row** for simplicity. ### Step 1: Write the formula for determinant Using cofactor expansion along the first row, the determinant is: $$\det(A) = 2 \cdot \det\left(\begin{bmatrix} 6 & 7 \\ 9 & 1 \end{bmatrix}\right) - 3 \cdot \det\left(\begin{bmatrix} 5 & 7 \\ 8 & 1 \end{bmatrix}\right) + 4 \cdot \det\left(\begin{bmatrix} 5 & 6 \\ 8 & 9 \end{bmatrix}\right).$$ --- ### Step 2: Compute minor determinants #### 1. Compute $$\det\left(\begin{bmatrix} 6 & 7 \\ 9 & 1 \end{bmatrix}\right)$$: $$\det = (6 \cdot 1) - (7 \cdot 9) = 6 - 63 = -57.$$ #### 2. Compute $$\det\left(\begin{bmatrix} 5 & 7 \\ 8 & 1 \end{bmatrix}\right)$$: $$\det = (5 \cdot 1) - (7 \cdot 8) = 5 - 56 = -51.$$ #### 3. Compute $$\det\left(\begin{bmatrix} 5 & 6 \\ 8 & 9 \end{bmatrix}\right)$$: $$\det = (5 \cdot 9) - (6 \cdot 8) = 45 - 48 = -3.$$ --- ### Step 3: Substitute into the formula $$\det(A) = 2 \cdot (-57) - 3 \cdot (-51) + 4 \cdot (-3).$$ Simplify each term: 1. $$2 \cdot (-57) = -114$$, 2. $$-3 \cdot (-51) = 153$$, 3. $$4 \cdot (-3) = -12$$. --- ### Step 4: Final computation $$\det(A) = -114 + 153 - 12 = 27.$$ --- ### Final Answer: $$\det(A) = 27.$$ Would you like detailed verification of each step or assistance with part 2? Here are some follow-up questions for deeper exploration: 1. Would you like to see cofactor expansion along a different row or column for comparison? 2. Do you need help with properties of determinants for simplifications? 3. Shall I explain why the determinant is significant in linear algebra? 4. Would you like a refresher on matrix row operations and their impact on determinants? 5. Do you need help interpreting the result of this determinant geometrically? **Tip**: Always check your arithmetic carefully when calculating determinants manually.Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Linear Algebra
Matrix Determinants
Formulas
Determinant of a 3x3 matrix using cofactor expansion: det(A) = a(ei − fh) − b(di − fg) + c(dh − eg)
Theorems
Cofactor Expansion Theorem
Suitable Grade Level
Grades 11-12, Undergraduate