In Exercises 23–30, use expansion by minors to evaluate each determinant. ∣310−340−13−5∣\begin{vmatrix} 3 & 1 & 0 \\ -3 & 4 & 0 \\ -1 & 3 & -5 \end{vmatrix}3−3−114300−5

Ch. 6 - Matrices and Determinants

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Ch. 6 - Matrices and Determinants

문제 25

7장, 문제 25

In Exercises 23–30, use expansion by minors to evaluate each determinant. $\begin{vmatrix}\) 3 & 1 & 0 \\ -3 & 4 & 0 \\ -1 & 3 & -5 \(\end{vmatrix}$

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Identify the 3x3 matrix for which you need to find the determinant using expansion by minors: \[\begin{vmatrix} 3 & 1 & 0 \\ -3 & 4 & 0 \\ -1 & 3 & -5 \end{vmatrix}\]

Choose a row or column to expand along. It is often easiest to choose a row or column with zeros to simplify calculations. Here, the third column has two zeros, so expand along the third column.

Write the determinant as a sum of the elements in the chosen column multiplied by their corresponding cofactors. For the third column, the determinant is: \[0 \cdot C_{13} + 0 \cdot C_{23} + (-5) \cdot C_{33}\] where $C_{ij}$ is the cofactor of the element in row $i$, column $j$.

Calculate the cofactor $C_{33}$. The cofactor is given by: \[C_{33} = (-1)^{3+3} \times M_{33}\] where $M_{33}$ is the minor obtained by deleting the third row and third column from the matrix. So, find the determinant of the 2x2 matrix: \[\begin{vmatrix} 3 & 1 \\ -3 & 4 \end{vmatrix}\]

Compute the 2x2 determinant: \[M_{33} = (3)(4) - (1)(-3)\] Then multiply by $(-1)^{6} = 1$ to get $C_{33}$. Finally, multiply $C_{33}$ by the element $-5$ from the original matrix to get the determinant of the 3x3 matrix.

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주요 개념

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Determinant of a Matrix

The determinant is a scalar value that can be computed from a square matrix and provides important properties such as invertibility. For a 3x3 matrix, the determinant helps determine if the matrix is singular or nonsingular and is essential in solving systems of linear equations.

Expansion by Minors

Expansion by minors is a method to calculate the determinant of a matrix by expanding along a row or column. It involves computing minors, which are determinants of smaller matrices formed by deleting one row and one column, and combining them with cofactors that include sign adjustments.

Cofactor and Sign Pattern

A cofactor is the minor of an element multiplied by (-1) raised to the sum of the element's row and column indices. The sign pattern alternates in a checkerboard fashion starting with a positive sign at the top-left element, which is crucial for correctly calculating the determinant using expansion by minors.

In Exercises 23–30, use expansion by minors to evaluate each determinant. ∣310−340−13−5∣\(\begin{vmatrix}\) 3 & 1 & 0 \\ -3 & 4 & 0 \\ -1 & 3 & -5 \(\end{vmatrix}\)3−3−114300−5