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Ch. 5 - Systems and Matrices
Lial - College Algebra 13th Edition
Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook
All textbooksLial 13th EditionCh. 5 - Systems and MatricesProblem 19
Chapter 6, Problem 19

Find the cofactor of each element in the second row of each matrix.
[12−123−2−141]\(\left\)[ \(\begin{matrix}\) 1 & 2 & -1 \\ 2 & 3 & -2 \\ -1 & 4 & 1 \(\end{matrix}\) \(\right\)]

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1
Identify the elements in the second row of the given matrix. Label them as \(a_{21}\), \(a_{22}\), \(a_{23}\), etc., depending on the size of the matrix.
For each element in the second row, find its minor. The minor of an element \(a_{ij}\) is the determinant of the matrix that remains after removing the \(i\)-th row and \(j\)-th column from the original matrix.
Calculate the determinant of each minor matrix. This involves using the standard determinant formulas depending on the size of the minor (2x2, 3x3, etc.).
Apply the cofactor formula for each element in the second row: \(C_{ij} = (-1)^{i+j} \times M_{ij}\), where \(M_{ij}\) is the minor of element \(a_{ij}\) and \(i\) and \(j\) are the row and column indices of the element.
Write down the cofactor for each element in the second row using the values obtained from the previous steps.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Matrix Minors

A minor of an element in a matrix is the determinant of the smaller matrix formed by deleting the row and column of that element. It is used as a building block to calculate cofactors and determinants.
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Cofactors

The cofactor of an element is the minor of that element multiplied by (-1) raised to the sum of the element's row and column indices. Cofactors are essential in calculating determinants and adjugates.

Row and Column Indexing in Matrices

Matrix elements are identified by their row and column positions, usually starting at 1. Understanding indexing is crucial to correctly find minors and cofactors, especially when focusing on a specific row.
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