Evaluate each determinant in Exercises 49–52. ∣428−7−20415005400−1∣\begin{vmatrix} 4 & 2 & 8 & -7 \\ -2 & 0 & 4 & 1 \\ 5 & 0 & 0 & 5 \\ 4 & 0 & 0 & -1 \end{vmatrix}4−25420008400−715−1

Ch. 6 - Matrices and Determinants

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Blitzer 8th Edition

Ch. 6 - Matrices and Determinants

문제 49

7장, 문제 49

Evaluate each determinant in Exercises 49–52. $\begin{vmatrix}\) 4 & 2 & 8 & -7 \\ -2 & 0 & 4 & 1 \\ 5 & 0 & 0 & 5 \\ 4 & 0 & 0 & -1 \(\end{vmatrix}$

검증된 단계별 안내

Step 1: Identify the matrix whose determinant you need to evaluate. The matrix is a 4x4 matrix: \[\begin{bmatrix} 4 & 2 & 8 & -7 \\ -2 & 0 & 4 & 1 \\ 5 & 0 & 0 & 5 \\ 4 & 0 & 0 & -1 \end{bmatrix}\]

Step 2: Choose a method to calculate the determinant of a 4x4 matrix. Common methods include expansion by minors (cofactor expansion) or using row operations to simplify the matrix to an upper triangular form.

Step 3: If using cofactor expansion, select a row or column with the most zeros to simplify calculations. In this matrix, the 3rd or 4th columns have zeros, so expanding along one of these columns can reduce the number of terms.

Step 4: Perform the cofactor expansion along the chosen column. For each element in that column, calculate its minor (the determinant of the 3x3 matrix that remains after removing the element's row and column) and multiply by the element and the appropriate sign (+ or -) based on its position.

Step 5: Calculate each 3x3 determinant using the standard formula or further expansion, then sum all the terms from the cofactor expansion to find the determinant of the original 4x4 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 a square matrix and provides important properties such as invertibility. For a 4x4 matrix, the determinant helps determine if the matrix is singular or nonsingular, which is crucial in solving systems of linear equations.

Expansion by Minors (Cofactor Expansion)

This method calculates the determinant of larger matrices by expanding along a row or column. It involves computing smaller determinants (minors) and applying alternating signs (cofactors), simplifying the calculation of a 4x4 determinant into manageable parts.

Properties of Determinants

Certain properties, such as the effect of row operations on the determinant, can simplify calculations. For example, swapping rows changes the sign, multiplying a row scales the determinant, and adding multiples of one row to another does not change the determinant, aiding efficient evaluation.

Evaluate each determinant in Exercises 49–52. ∣428−7−20415005400−1∣\(\begin{vmatrix}\) 4 & 2 & 8 & -7 \\ -2 & 0 & 4 & 1 \\ 5 & 0 & 0 & 5 \\ 4 & 0 & 0 & -1 \(\end{vmatrix}\)4−25420008400−715−1