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

Guida verificata passo dopo passo
1
Identify the size of the matrix for which you need to evaluate the determinant (e.g., 2x2, 3x3, etc.).
For a 2x2 matrix \( \begin{bmatrix} a & b \\ c & d \end{bmatrix} \), use the formula for the determinant: \( \det = ad - bc \).
For a 3x3 matrix \( \begin{bmatrix} a & b & c \\ d & e & f \\ g & h & i \end{bmatrix} \), apply the rule of Sarrus or cofactor expansion to find the determinant.
If using cofactor expansion, select a row or column (usually one with zeros to simplify calculations), then calculate the minors and cofactors for each element in that row or column.
Sum the products of each element and its corresponding cofactor to get the determinant: \( \det = a_{ij}C_{ij} + a_{ik}C_{ik} + \ldots \), where \( C_{ij} \) are the cofactors.

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Determinant of a Matrix

The determinant is a scalar value computed from a square matrix that provides important properties about the matrix, such as invertibility. It is calculated using specific rules depending on the matrix size, like expansion by minors or row operations.
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Determinants of 2×2 Matrices

Methods for Evaluating Determinants

Determinants can be evaluated using various methods including cofactor expansion, row reduction to echelon form, or special shortcuts for certain matrix types. Choosing an efficient method simplifies the calculation process.
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Determinants of 2×2 Matrices

Properties of Determinants

Determinants have properties such as linearity, the effect of row swaps (which change the sign), and the determinant of a product equals the product of determinants. Understanding these helps in simplifying and verifying determinant calculations.
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Determinants of 2×2 Matrices
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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 \(\end{vmatrix}\)

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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 \(\end{vmatrix}\)

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Domanda del libro di testo

In Exercises 46–51, evaluate each determinant.

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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.

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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.

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Domanda del libro di testo

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 \(\end{vmatrix}\)

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