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5 1 -2 0 3 0 1 -2 0 4 3 1

5 1 -2 0 3 0 1 -2 0 4 3 1

Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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**Finding the Determinant of a 4x4 Matrix** **Problem Statement:** Show carefully the complete work (no shortcuts, no technology) in the process for finding the determinant. Give the determinant. **Matrix:** \[ \begin{bmatrix} 5 & 0 & 0 & 1 \\ -2 & 0 & 3 & 0 \\ 0 & 1 & -2 & 0 \\ 4 & 3 & 0 & 1 \\ \end{bmatrix} \] **Step-by-Step Process:** 1. **Choose a Row or Column:** - Typically, select a row or column with the most zeros for easier calculation. In this matrix, row 1 has two zeros. 2. **Cofactor Expansion along Row 1:** - Calculate the determinant using the formula: \( \det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} + a_{14}C_{14} \). 3. **Cofactors Calculation:** - \( C_{11} = \det \begin{bmatrix} 0 & 3 & 0 \\ 1 & -2 & 0 \\ 3 & 0 & 1 \\ \end{bmatrix} \) - \( C_{14} = \det \begin{bmatrix} -2 & 0 & 3 \\ 0 & 1 & -2 \\ 4 & 3 & 0 \\ \end{bmatrix} \) 4. **Calculate the 3x3 Determinants:** - Use cofactor expansion again for each 3x3 matrix. 5. **Apply Signs Based on Position:** - In a 4x4 matrix, alternate signs starting with positive for element \( a_{11} \). 6. **Combine Results:** - Compute the determinant by considering each product of elements and their respective cofactors. 7. **Final Answer:** - Present the determinant value of the matrix after all calculations. **Note:** Detailed calculations for each step are fundamental to understanding matrix determinant computation and verifying results without using technology.
Transcribed Image Text:**Finding the Determinant of a 4x4 Matrix** **Problem Statement:** Show carefully the complete work (no shortcuts, no technology) in the process for finding the determinant. Give the determinant. **Matrix:** \[ \begin{bmatrix} 5 & 0 & 0 & 1 \\ -2 & 0 & 3 & 0 \\ 0 & 1 & -2 & 0 \\ 4 & 3 & 0 & 1 \\ \end{bmatrix} \] **Step-by-Step Process:** 1. **Choose a Row or Column:** - Typically, select a row or column with the most zeros for easier calculation. In this matrix, row 1 has two zeros. 2. **Cofactor Expansion along Row 1:** - Calculate the determinant using the formula: \( \det(A) = a_{11}C_{11} + a_{12}C_{12} + a_{13}C_{13} + a_{14}C_{14} \). 3. **Cofactors Calculation:** - \( C_{11} = \det \begin{bmatrix} 0 & 3 & 0 \\ 1 & -2 & 0 \\ 3 & 0 & 1 \\ \end{bmatrix} \) - \( C_{14} = \det \begin{bmatrix} -2 & 0 & 3 \\ 0 & 1 & -2 \\ 4 & 3 & 0 \\ \end{bmatrix} \) 4. **Calculate the 3x3 Determinants:** - Use cofactor expansion again for each 3x3 matrix. 5. **Apply Signs Based on Position:** - In a 4x4 matrix, alternate signs starting with positive for element \( a_{11} \). 6. **Combine Results:** - Compute the determinant by considering each product of elements and their respective cofactors. 7. **Final Answer:** - Present the determinant value of the matrix after all calculations. **Note:** Detailed calculations for each step are fundamental to understanding matrix determinant computation and verifying results without using technology.
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