Find the cofactor of each element in the second row of each matrix. [−201120421]\left[ \begin{matrix} -2 & 0 & 1 \\ 1 & 2 & 0 \\ 4 & 2 & 1 \end{matrix} \right]

Ch. 5 - Systems and Matrices

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Lial 13th Edition

Ch. 5 - Systems and Matrices

문제 17

6장, 문제 17

Find the cofactor of each element in the second row of each matrix.
$[\begin{matrix} - 2 & 0 & 1 \\ 1 & 2 & 0 \\ 4 & 2 & 1 \end{matrix}]$

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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 applying the determinant formula appropriate for the size of the minor (e.g., 2x2 or 3x3).

Determine the cofactor for each element in the second row by applying the sign factor $(-1)^{i+j}$ to the minor. For an element in row 2 and column $j$, the cofactor is $C_{2j} = (-1)^{2+j} \times M_{2j}$, where $M_{2j}$ is the minor determinant.

Write down the cofactor values for each element in the second row, which are the signed minors calculated in the previous step.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

Cofactors

The cofactor of an element in a matrix 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 how to locate elements by their indices is crucial for finding minors and cofactors, especially when focusing on a specific row.

Find the cofactor of each element in the second row of each matrix. [−201120421]\(\left\)[ \(\begin{matrix}\) -2 & 0 & 1 \\ 1 & 2 & 0 \\ 4 & 2 & 1 \(\end{matrix}\) \(\right\)]