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Ch. 6 - Matrices and Determinants
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 6 - Matrices and DeterminantsProblem 13
Chapter 7, Problem 13

In Exercises 1 - 24, use Gaussian Elimination to find the complete solution to each system of equations, or show that none exists. {w−3x+y−4z=4−2w+x+2y=−23w−2x+y−6z=2−w+3x+2y−z=−6\(\begin{cases}\)w - 3x + y - 4z = 4 \\-2w + x + 2y = -2 \\3w - 2x + y - 6z = 2 \\-w + 3x + 2y - z = -6\(\end{cases}\)

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Write the system of equations as an augmented matrix. Each row corresponds to an equation, and each column corresponds to the coefficients of variables w, x, y, z, and the constants on the right side. The matrix looks like this: \[\begin{bmatrix} 1 & -3 & 1 & -4 & | & 4 \\ -2 & 1 & 2 & 0 & | & -2 \\ 3 & -2 & 1 & -6 & | & 2 \\ -1 & 3 & 2 & -1 & | & -6 \end{bmatrix}\]
Use row operations to create zeros below the leading 1 in the first column (the pivot position). For example, add 2 times row 1 to row 2, subtract 3 times row 1 from row 3, and add row 1 to row 4. This step transforms the matrix to have zeros below the first pivot.
Move to the second column and create a leading 1 in the second row, second column if necessary (by scaling the row), then use row operations to create zeros above and below this pivot. This process continues the elimination to get the matrix closer to row echelon form.
Repeat the process for the third and fourth columns: create leading 1s (pivots) and use row operations to eliminate other entries in those columns, aiming for reduced row echelon form where each pivot is the only nonzero entry in its column.
Once the matrix is in reduced row echelon form, translate the matrix back into equations. From these equations, express the variables in terms of any free variables if they exist, or find the unique solution if the system is consistent and determined. If a contradiction appears (like 0 = nonzero), conclude that no solution exists.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Systems of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. The goal is to find values for these 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 solve by back-substitution or to determine if no solution exists.
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Row Operations and Consistency of Systems

Row operations include swapping rows, multiplying a row by a nonzero scalar, and adding multiples of one row to another. These operations preserve the solution set and help identify if a system is consistent (has solutions) or inconsistent (no solutions). Recognizing inconsistent rows is key to concluding no solution exists.
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Related Practice
Textbook Question

Find the following matrices: A+BA + B

A=[2−41],B=[−53−1]A = \(\begin{bmatrix}\)2 \\-4 \\1\(\end{bmatrix}\), B = \(\begin{bmatrix}\)-5 \\3 \\-1\(\end{bmatrix}\)

Textbook Question

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=[23−12]A = \(\begin{bmatrix}\) 2 & 3 \\ -1 & 2 \(\end{bmatrix}\)

Textbook Question

For Exercises 11–22, use Cramer's Rule to solve each system. {12x+3y=152x−3y=13\(\begin{cases}\)12x + 3y = 15 \\2x - 3y = 13\(\end{cases}\)

Textbook Question

In Exercises 9 - 16, find the following matrices: b. A - B

Textbook Question

In Exercises 9 - 16, find the following matrices: c. - 4A

Textbook Question

Perform each matrix row operation and write the new matrix.

[2−64∣1015−5∣0304∣7]12R1\(\begin{bmatrix}\)2 & -6 & 4 & \(\vert\) & 10 \\1 & 5 & -5 & \(\vert\) & 0 \\3 & 0 & 4 & \(\vert\) & 7\(\end{bmatrix}\]\quad\) \(\frac{1}{2}\)R_1

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