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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
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Tutti i libri di testoBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblema 39
Capitolo 7, Problema 39

In Exercises 37–44, use Cramer's Rule to solve each system. {4x−5y−6z=−1x−2y−5z=−122x−y=7\(\begin{cases}\) 4x - 5y - 6z = -1 \\ x - 2y - 5z = -12 \\ 2x - y = 7 \(\end{cases}\)

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Write the system of equations in standard form, ensuring all variables are on the left side and constants on the right side: \(\begin{cases} 4x - 5y - 6z = -1 \\ x - 2y - 5z = -12 \\ 2x - y + 0z = 7 \end{cases}\)
Form the coefficient matrix \(A\) from the coefficients of \(x\), \(y\), and \(z\) in the system: \(A = \begin{bmatrix} 4 & -5 & -6 \\ 1 & -2 & -5 \\ 2 & -1 & 0 \end{bmatrix}\)
Calculate the determinant of matrix \(A\), denoted as \(\det(A)\), which is necessary to apply Cramer's Rule. This involves expanding the determinant using minors and cofactors.
Form matrices \(A_x\), \(A_y\), and \(A_z\) by replacing the respective columns of \(A\) with the constants vector \(\mathbf{b} = \begin{bmatrix} -1 \\ -12 \\ 7 \end{bmatrix}\): - \(A_x\) is formed by replacing the first column of \(A\) with \(\mathbf{b}\). - \(A_y\) is formed by replacing the second column of \(A\) with \(\mathbf{b}\). - \(A_z\) is formed by replacing the third column of \(A\) with \(\mathbf{b}\).
Calculate the determinants \(\det(A_x)\), \(\det(A_y)\), and \(\det(A_z)\). Then, use Cramer's Rule to find the solutions: \(x = \frac{\det(A_x)}{\det(A)}\), \(y = \frac{\det(A_y)}{\det(A)}\), \(z = \frac{\det(A_z)}{\det(A)}\).

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Cramer's Rule

Cramer's Rule is a method for solving systems of linear equations using determinants. It applies when the system has the same number of equations as unknowns and the coefficient matrix has a non-zero determinant. Each variable is found by replacing the corresponding column in the coefficient matrix with the constants vector and calculating the determinant ratio.
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Cramer's Rule - 2 Equations with 2 Unknowns

Determinants of Matrices

The determinant is a scalar value that can be computed from a square matrix and provides important properties about the matrix, such as invertibility. For a 3x3 matrix, the determinant is calculated using a specific formula involving minors and cofactors. A non-zero determinant indicates the system has a unique solution.
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Determinants of 2×2 Matrices

Systems of Linear Equations

A system of linear equations consists of multiple linear equations with the same variables. Solving the system means finding values for the variables that satisfy all equations simultaneously. Methods include substitution, elimination, matrix operations, and Cramer's Rule, especially useful for small systems.
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Introduction to Systems of Linear Equations
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Solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

{3w−4x+y+z=9w+x−y−z=02w+x+4y−2z=3−w+2x+y−3z=3\(\begin{cases}\) 3w - 4x + y + z = 9 \\ w + x - y - z = 0 \\ 2w + x + 4y - 2z = 3 \\ -w + 2x + y - 3z = 3 \(\end{cases}\)

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a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.

{x−y+z=82y−z=−72x+3y=1The inverse of [1−1102−1230] is [33−1−2−21−4−52].\(\begin{cases}\) x - y + z = 8 \\ 2y - z = -7 \\ 2x + 3y = 1 \(\end{cases}\) \\ \(\text{The inverse of }\) \(\begin{bmatrix}\) 1 & -1 & 1 \\ 0 & 2 & -1 \\ 2 & 3 & 0 \(\end{bmatrix}\) \(\text{ is }\) \(\begin{bmatrix}\) 3 & 3 & -1 \\ -2 & -2 & 1 \\ -4 & -5 & 2 \(\end{bmatrix}\)\(\text{.}\)

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a. Write each linear system as a matrix equation in the form AX = B. b. Solve the system using the inverse that is given for the coefficient matrix.

{2x+6y+6z=82x+7y+6z=102x+7y+7z=9The inverse of [266276277] is [720−3−1000−11].\(\begin{cases}\) 2x + 6y + 6z = 8 \\ 2x + 7y + 6z = 10 \\ 2x + 7y + 7z = 9 \(\end{cases}\) \\ \(\text{The inverse of }\) \(\begin{bmatrix}\) 2 & 6 & 6 \\ 2 & 7 & 6 \\ 2 & 7 & 7 \(\end{bmatrix}\) \(\text{ is }\) \(\begin{bmatrix}\) \(\frac{7}{2}\) & 0 & -3 \\ -1 & 0 & 0 \\ 0 & -1 & 1 \(\end{bmatrix}\)\(\text{.}\)

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In Exercises 39–42, find A^(-1) Check that AA^-1 = I and A^(-1)A = I

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Perform the indicated matrix operations given that A, B and C are defined as follows. If an operation is not defined, state the reason.

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Find the quadratic function f(x) = ax² + bx + c for which ƒ( − 2) = −4, ƒ(1) = 2, and f(2) = 0.

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