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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 33
Capitolo 7, Problema 33

Write each matrix equation as a system of linear equations without matrices.
[4−72−3][xy]=[−31]\(\begin{bmatrix}\) 4 & -7 \\ 2 & -3 \(\end{bmatrix}\) \(\begin{bmatrix}\) x \\ y \(\end{bmatrix}\) = \(\begin{bmatrix}\) -3 \\ 1 \(\end{bmatrix}\)

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Identify the given matrix equation: \(\left[ \begin{array}{cc} 4 & -7 \\ 2 & -3 \end{array} \right] \left[ \begin{array}{c} x \\ y \end{array} \right] = \left[ \begin{array}{c} -3 \\ 1 \end{array} \right]\).
Recall that multiplying a matrix by a vector corresponds to forming linear combinations of the vector components with the matrix rows.
Write the first row multiplication as an equation: \(4x - 7y = -3\).
Write the second row multiplication as an equation: \(2x - 3y = 1\).
Thus, the matrix equation is equivalent to the system of linear equations: \(\begin{cases} 4x - 7y = -3 \\ 2x - 3y = 1 \end{cases}\).

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Matrix Multiplication

Matrix multiplication involves multiplying rows of the first matrix by columns of the second matrix and summing the products. In this problem, multiplying the 2x2 coefficient matrix by the 2x1 variable matrix results in a 2x1 matrix representing the system's left side.
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System of Linear Equations

A system of linear equations consists of multiple linear equations with the same variables. Writing the matrix equation as a system means expressing each row multiplication as an individual linear equation involving variables x and y.
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Introduction to Systems of Linear Equations

Equating Matrices to Form Equations

When two matrices are equal, their corresponding entries are equal. This principle allows us to set each element of the product matrix equal to the corresponding element in the constant matrix, forming a system of equations to solve.
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Solving Systems of Equations - Matrices (Reduced Row-Echelon Form)
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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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Domanda del libro di testo

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

{2x+2y+7z=−12x+y+2z=24x+6y+z=15\(\begin{cases}\) 2x + 2y + 7z = -1 \\ 2x + y + 2z = 2 \\ 4x + 6y + z = 15 \(\end{cases}\)

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Domanda del libro di testo

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

{w+x+y+z=42w+x−2y−z=0w−2x−y−2z=−23w+2x+y+3z=4\(\begin{cases}\) w + x + y + z = 4 \\ 2w + x - 2y - z = 0 \\ w - 2x - y - 2z = -2 \\ 3w + 2x + y + 3z = 4 \(\end{cases}\)

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Domanda del libro di testo

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 1 - 1 4 1 1 0 A = 4 - 1 3 B = 1 2 4 2 0 - 2 1 - 1 3

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Domanda del libro di testo

In Exercises 27 - 36, find (if possible) the following matrices: a. AB b. BA 4 2 2 3 4 A = 6 1 B = 3 5 - 1 - 2 0

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Domanda del libro di testo

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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