Skip to main content
Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblema 49
Capitolo 7, Problema 49

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 \(\end{vmatrix}\)

Guida verificata passo dopo passo
1
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.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
7m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

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.
Video consigliato:
4:36
Determinants of 2×2 Matrices

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.
Video consigliato:
5:12
Graph Ellipses at Origin

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.
Video consigliato:
4:36
Determinants of 2×2 Matrices
Pratica correlata
Domanda del libro di testo

Evaluate each determinant.

∣2222022200220002∣\(\begin{vmatrix}\) 2 & 2 & 2 & 2 \\ 0 & 2 & 2 & 2 \\ 0 & 0 & 2 & 2 \\ 0 & 0 & 0 & 2 \(\end{vmatrix}\)

793
views
Domanda del libro di testo

Evaluate each determinant in Exercises 49–52. ∣−2−3351−400122−32011∣\(\begin{vmatrix}\) -2 & -3 & 3 & 5 \\ 1 & -4 & 0 & 0 \\ 1 & 2 & 2 & -3 \\ 2 & 0 & 1 & 1 \(\end{vmatrix}\)

811
views
Domanda del libro di testo

In Exercises 46–51, evaluate each determinant.

985
views
Domanda del libro di testo

In Exercises 45–48, explain why the system of equations cannot be solved using Cramer's Rule. Then use Gaussian elimination to solve the system.

854
views
Domanda del libro di testo

In Exercises 53–54, evaluate each determinant. ∣∣31−23∣∣7015∣∣3007∣∣9−635∣∣\(\begin{vmatrix}\) \(\begin{vmatrix}\) 3 & 1 \\ -2 & 3 \(\end{vmatrix}\) & \(\begin{vmatrix}\) 7 & 0 \\ 1 & 5 \(\end{vmatrix}\) \\ \(\begin{vmatrix}\) 3 & 0 \\ 0 & 7 \(\end{vmatrix}\) & \(\begin{vmatrix}\) 9 & -6 \\ 3 & 5 \(\end{vmatrix}\) \(\end{vmatrix}\)

801
views
Domanda del libro di testo

Evaluate each determinant.

∣470−56032−4∣\(\begin{vmatrix}\) 4 & 7 & 0 \\ -5 & 6 & 0 \\ 3 & 2 & -4 \(\end{vmatrix}\)

778
views