Evaluate by constructing zeros in a column or in a row. Do not forget to factor out common factors from rows and columns. (a) 1 1 1 2 1 1 4 1392 1 4 16 8 27 64 (b) 4 6732 7 7 0 3 2 18 14 0 1 27 0 45 5 2 сл с ся сл сл 5 -2 (c) 5 5 5 2 -2 12481 2 16 1 3 9 27 16 81 ܝܕ ܒ ܥ ܗܘ ܗ -2 -8
Evaluate by constructing zeros in a column or in a row. Do not forget to factor out common factors from rows and columns. (a) 1 1 1 2 1 1 4 1392 1 4 16 8 27 64 (b) 4 6732 7 7 0 3 2 18 14 0 1 27 0 45 5 2 сл с ся сл сл 5 -2 (c) 5 5 5 2 -2 12481 2 16 1 3 9 27 16 81 ܝܕ ܒ ܥ ܗܘ ܗ -2 -8
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Matrix Evaluation and Factorization**
**Instruction:**
Evaluate by constructing zeros in a column or in a row. Do not forget to factor out common factors from rows and columns.
---
**Matrices:**
(a)
\[
\begin{pmatrix}
1 & 1 & 1 \\
1 & 2 & 3 \\
4 & 9 & 16 \\
8 & 27 & 64 \\
\end{pmatrix}
\]
(b)
\[
\begin{pmatrix}
4 & 6 & 2 & 18 \\
0 & 7 & 7 & 14 \\
0 & 3 & 1 & 27 \\
3 & 2 & 0 & 45 \\
\end{pmatrix}
\]
(c)
\[
\begin{pmatrix}
5 & 5 & 5 & 2 & 1 \\
-2 & 2 & 2 & -2 & 3 \\
-2 & 4 & 4 & -4 & 9 \\
-8 & 8 & 8 & 8 & 27 \\
2 & 16 & 16 & 16 & 81 \\
\end{pmatrix}
\]
**Guidelines:**
1. **Identify Common Factors:**
- Check each row and column for common factors. Extract and simplify the matrices by factoring these out.
2. **Constructing Zeros:**
- Use elementary row or column operations to create zeros in strategic positions, which will simplify the process of evaluating determinants.
3. **Simplification:**
- Continue to simplify until the matrices are in their simplest form for evaluation. Each zero constructed makes the evaluation of determinants easier.
**Tips:**
- Look closely for patterns or numerical sequences within rows or columns.
- Ensure that by factoring out, the determinant’s properties remain unchanged.
This exercise enhances understanding of matrix operations and determinants, critical in solving linear algebra problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F801d34df-dcab-45aa-85e0-2349cce424a4%2F65bd22ec-45d6-4f41-b789-b5380e80a5b1%2Fuy1vsz_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Matrix Evaluation and Factorization**
**Instruction:**
Evaluate by constructing zeros in a column or in a row. Do not forget to factor out common factors from rows and columns.
---
**Matrices:**
(a)
\[
\begin{pmatrix}
1 & 1 & 1 \\
1 & 2 & 3 \\
4 & 9 & 16 \\
8 & 27 & 64 \\
\end{pmatrix}
\]
(b)
\[
\begin{pmatrix}
4 & 6 & 2 & 18 \\
0 & 7 & 7 & 14 \\
0 & 3 & 1 & 27 \\
3 & 2 & 0 & 45 \\
\end{pmatrix}
\]
(c)
\[
\begin{pmatrix}
5 & 5 & 5 & 2 & 1 \\
-2 & 2 & 2 & -2 & 3 \\
-2 & 4 & 4 & -4 & 9 \\
-8 & 8 & 8 & 8 & 27 \\
2 & 16 & 16 & 16 & 81 \\
\end{pmatrix}
\]
**Guidelines:**
1. **Identify Common Factors:**
- Check each row and column for common factors. Extract and simplify the matrices by factoring these out.
2. **Constructing Zeros:**
- Use elementary row or column operations to create zeros in strategic positions, which will simplify the process of evaluating determinants.
3. **Simplification:**
- Continue to simplify until the matrices are in their simplest form for evaluation. Each zero constructed makes the evaluation of determinants easier.
**Tips:**
- Look closely for patterns or numerical sequences within rows or columns.
- Ensure that by factoring out, the determinant’s properties remain unchanged.
This exercise enhances understanding of matrix operations and determinants, critical in solving linear algebra problems.
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