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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 31
Capitolo 7, Problema 31

In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. ∣−34−55−208−13∣\(\begin{vmatrix}\) -3 & 4 & -5 \\ 5 & -2 & 0 \\ 8 & -1 & 3 \(\end{vmatrix}\)

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Write down the determinant matrix as: \[\begin{vmatrix} -3 & 4 & -5 \\ 5 & -2 & 0 \\ 8 & -1 & 3 \end{vmatrix}\]
Use the alternative method (also known as the rule of Sarrus) for evaluating a 3x3 determinant. First, rewrite the first two columns of the matrix to the right of the original matrix: \[\begin{matrix} -3 & 4 & -5 & | & -3 & 4 \\ 5 & -2 & 0 & | & 5 & -2 \\ 8 & -1 & 3 & | & 8 & -1 \end{matrix}\]
Calculate the sum of the products of the diagonals going from top-left to bottom-right: \[(-3) \times (-2) \times 3 + 4 \times 0 \times 8 + (-5) \times 5 \times (-1)\]
Calculate the sum of the products of the diagonals going from bottom-left to top-right: \[8 \times (-2) \times (-5) + (-1) \times 0 \times (-3) + 3 \times 5 \times 4\]
Find the determinant by subtracting the second sum from the first sum: \[\text{Determinant} = \left[(-3)(-2)(3) + 4(0)(8) + (-5)(5)(-1)\right] - \left[8(-2)(-5) + (-1)(0)(-3) + 3(5)(4)\right]\]

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Third-Order Determinants

A third-order determinant is a scalar value calculated from a 3x3 matrix. It helps determine properties like matrix invertibility and solutions to systems of equations. The determinant is computed using specific methods such as expansion by minors or the alternative method.
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Determinants of 3×3 Matrices

Alternative Method for Evaluating Determinants

The alternative method, often called the diagonal or Sarrus' rule, is a shortcut for calculating 3x3 determinants. It involves summing the products of diagonals from left to right and subtracting the products of diagonals from right to left, simplifying the calculation process.
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Determinants of 3×3 Matrices

Properties of Determinants in Systems of Equations

Determinants are used to analyze systems of linear equations; a nonzero determinant indicates a unique solution. Understanding how to evaluate determinants helps in solving systems using Cramer's rule and assessing matrix invertibility.
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Introduction to Systems of Linear Equations
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Write each linear system as a matrix equation in the form AX = B, where A is the coefficient matrix and B is the constant matrix.

{x+3y+4z=−3x+2y+3z=−2x+4y+3z=−6\(\begin{cases}\) x + 3y + 4z = -3 \\ x + 2y + 3z = -2 \\ x + 4y + 3z = -6 \(\end{cases}\)

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In Exercises 31–36, use the alternative method for evaluating third-order determinants on here to evaluate each determinant. ∣1561451910∣\(\begin{vmatrix}\) 1 & 5 & 6 \\ 1 & 4 & 5 \\ 1 & 9 & 10 \(\end{vmatrix}\)

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Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

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Write each matrix equation as a system of linear equations without matrices.

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