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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 17
Capitolo 7, Problema 17

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {x+2y+3z=5y−5z=0\(\begin{cases}\) x + 2y + 3z = 5 \\ y - 5z = 0 \(\end{cases}\)

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Write the system of equations in augmented matrix form. For the system: \(x + 2y + 3z = 5\) \(0x + y - 5z = 0\), the augmented matrix is: \[\left[\begin{array}{ccc|c} 1 & 2 & 3 & 5 \\ 0 & 1 & -5 & 0 \end{array}\right]\]
Use Gaussian elimination to get the matrix into row-echelon form. The first row already has a leading 1 in the first column. The second row has a leading 1 in the second column, so the matrix is already in row-echelon form.
Express the system back into equations from the row-echelon form: Row 1: \(x + 2y + 3z = 5\) Row 2: \(y - 5z = 0\)
From the second equation, solve for \(y\) in terms of \(z\): \(y = 5z\)
Substitute \(y = 5z\) into the first equation to express \(x\) in terms of \(z\): \(x + 2(5z) + 3z = 5\) Simplify and solve for \(x\) in terms of \(z\).

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Systems of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The goal is to find values for the variables that satisfy all equations simultaneously. Understanding how to represent and interpret these systems is fundamental before applying solution methods.
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Introduction to Systems of Linear Equations

Gaussian Elimination

Gaussian elimination is a systematic method for solving systems of linear equations by transforming the system's augmented matrix into row-echelon form using row operations. This process simplifies the system, making it easier to find solutions or determine if none exist.
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Solving Systems of Equations - Elimination

Row Operations and Row-Echelon Form

Row operations include swapping rows, multiplying a row by a nonzero scalar, and adding multiples of one row to another. These operations are used to convert the augmented matrix into row-echelon form, where the system can be solved by back-substitution to find the complete solution set.
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Performing Row Operations on Matrices
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Use the fact that if A=[abcd]A=\(\begin{bmatrix}\)a & b\\ c & d\(\end{bmatrix}\), then A−1=1ad−bc[d−b−ca]A^{-1}=\(\frac{1}{ad-bc}\)\(\begin{bmatrix}\)d & -b\\ -c & a\(\end{bmatrix}\) to find the inverse of each matrix, if possible. Check that AA−1=I2AA^{-1} = I_2 and A−1A=I2A^{-1}A = I_2.

A=[10−2−51]A = \(\begin{bmatrix}\) 10 & -2 \\ -5 & 1 \(\end{bmatrix}\)

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In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {x+y−2z=23x−y−6z=−7\(\begin{cases}\) x + y - 2z = 2 \\ 3x - y - 6z = -7 \(\end{cases}\)

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Let A=[−3−72−950]A = \(\begin{bmatrix}\) -3 & -7 \\ 2 & -9 \\ 5 & 0 \(\end{bmatrix}\) and B=[−5−1003−4]B = \(\begin{bmatrix}\) -5 & -1 \\ 0 & 0 \\ 3 & -4 \(\end{bmatrix}\) Solve each matrix equation for X. X - A = B

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In Exercises 14–27, perform the indicated matrix operations given that and D are defined as follows. If an operation is not defined, state the reason. 3A+2D

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Domanda del libro di testo

In Exercises 19–20, a few steps in the process of simplifying the given matrix to row-echelon form, with 1s down the diagonal from upper left to lower right, and 0s below the 1s, are shown. Fill in the missing numbers in the steps that are shown.

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Domanda del libro di testo

Perform each matrix row operation and write the new matrix.

[1−111∣301−2−1∣02034∣115124∣6]−2R1+R3−5R1+R4\(\begin{bmatrix}\) 1 & -1 & 1 & 1 & \(\vert\) & 3 \\ 0 & 1 & -2 & -1 & \(\vert\) & 0 \\ 2 & 0 & 3 & 4 & \(\vert\) & 11 \\ 5 & 1 & 2 & 4 & \(\vert\) & 6 \(\end{bmatrix}\) \(\quad\) \(\begin{array}{l}\) -2R_1 + R_3 \\ -5R_1 + R_4 \(\end{array}\)

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