In Exercises 23–30, use expansion by minors to evaluate each determinant. ∣30021−525−1∣\begin{vmatrix} 3 & 0 & 0 \\ 2 & 1 & -5 \\ 2 & 5 & -1 \end{vmatrix}3220150−5−1

Ch. 6 - Matrices and Determinants

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

문제 23

7장, 문제 23

In Exercises 23–30, use expansion by minors to evaluate each determinant. $\begin{vmatrix}\) 3 & 0 & 0 \\ 2 & 1 & -5 \\ 2 & 5 & -1 \(\end{vmatrix}$

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Identify the matrix for which you need to find the determinant: \[\begin{bmatrix} 3 & 0 & 0 \\ 2 & 1 & -5 \\ 2 & 5 & -1 \end{bmatrix}\]

Choose a row or column to expand by minors. Since the first row has two zeros, expanding along the first row is efficient.

Write the determinant expansion along the first row: \[\text{det} = 3 \cdot C_{11} + 0 \cdot C_{12} + 0 \cdot C_{13}\] where $C_{ij}$ is the cofactor of the element in row $i$, column $j$.

Calculate the cofactor $C_{11}$ by finding the determinant of the 2x2 submatrix obtained by removing the first row and first column: \[\begin{bmatrix} 1 & -5 \\ 5 & -1 \end{bmatrix}\] The determinant of this submatrix is calculated as $(1)(-1) - (-5)(5)$.

Multiply the element $3$ by the cofactor $C_{11}$ to get the determinant of the original 3x3 matrix.

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주요 개념

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

Determinant of a Matrix

The determinant is a scalar value that can be computed from the elements of a square matrix. It provides important properties of the matrix, such as invertibility, and is used in solving systems of linear equations. For a 3x3 matrix, the determinant can be found using expansion by minors or other methods.

Expansion by Minors

Expansion by minors is a method to calculate the determinant of a matrix by breaking it down into smaller determinants of submatrices. This involves selecting a row or column, multiplying each element by the determinant of its minor matrix, and applying alternating signs. It simplifies the calculation of larger determinants.

Minor and Cofactor

A minor is the determinant of the smaller matrix formed by deleting one row and one column from the original matrix. The cofactor is the minor multiplied by (-1)^(row+column), which accounts for sign changes in expansion by minors. Understanding minors and cofactors is essential for correctly applying expansion by minors.

In Exercises 23–30, use expansion by minors to evaluate each determinant. ∣30021−525−1∣\(\begin{vmatrix}\) 3 & 0 & 0 \\ 2 & 1 & -5 \\ 2 & 5 & -1 \(\end{vmatrix}\)3220150−5−1