7. Systems of Equations & Matrices

Introduction to Matrices

College Algebra

7. Systems of Equations & Matrices

Introduction to Matrices

3:15 minutes

Problem 9aBlitzer - 8th Edition

Textbook Question

In Exercises 9-12, write the system of linear equations represented by the augmented matrix. Use x, y, and z, or, if necessary, w, x, y, and z, for the variables.

Verified step by step guidance

Identify the augmented matrix:

[\begin{matrix} 5 & 0 & 3 & | & - 11 \\ 0 & 1 & - 4 & | & 12 \\ 7 & 2 & 0 & | & 3 \end{matrix}]

Recognize that each row of the matrix corresponds to a linear equation.

Assign variables to the columns: let the first column be

x

, the second column be

y

, and the third column be

z

Write the first equation from the first row:

5 x + 0 y + 3 z = - 11

Write the second equation from the second row:

0 x + 1 y - 4 z = 12

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In Exercises 1–8, write the augmented matrix for each system of linear equations.

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In Exercises 1–2, perform each matrix row operation and write the new matrix.

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In Exercises 1 - 4, a. Give the order of each matrix, b. If A = [a_ij], identify a_32 and a_23, or explain why identification is not possible, 4 - 7 5 - 6 8 - 1 (please enclose the values above in a matrix symbol)

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In Exercises 3–5, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

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What is the augmented matrix of the following system? -3x + 5y = 2 6x + 2y = 7

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In Exercises 3–5, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

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Textbook Question

In Exercises 3–5, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

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Textbook Question

In Exercises 1–8, write the augmented matrix for each system of linear equations.

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Textbook Question

Use the given row transformation to change each matrix as indicated. See Sample 1. < 2x2 Matrix > ; -4 times row 1 added to row 2

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Textbook Question

In Exercises 1–8, write the augmented matrix for each system of linear equations.

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Textbook Question

Use the given row transformation to change each matrix as indicated. See Sample 1. < 2x2 Matrix > ; -7 times row 1 added to row 2

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Use the given row transformation to change each matrix as indicated. See Sample 1. < 3x3 Matrix > ; 2 times row 1 added to row 2

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Textbook Question

In Exercises 9 - 16, find the following matrices: b. A - B 4 1 5 9 A = B = 3 2 0 7

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Textbook Question

Find the dimension of each matrix. Identify any square, column, or row matrices. See the discussion preceding Example 1.

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In Exercises 9 - 16, find the following matrices: d. - 3A + 2B 4 1 5 9 A = B = 3 2 0 7

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Use the given row transformation to change each matrix as indicated. See Sample 1. < 3x3 Matrix > ; 4 times row 1 added to row 2

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Textbook Question

In Exercises 9-12, write the system of linear equations represented by the augmented matrix. Use x, y, and z, or, if necessary, w, x, y, and z, for the variables.

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In Exercises 9 - 16, find the following matrices: a. A + B 3 1 1 2 - 3 6 A = B = - 1 2 5 - 3 1 - 4

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Textbook Question

In Exercises 9 - 16, find the following matrices: d. - 3A + 2B 3 1 1 2 - 3 6 A = B = - 1 2 5 - 3 1 - 4

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In Exercises 9 - 16, find the following matrices: c. - 4A 3 1 1 2 - 3 6 A = B = - 1 2 5 - 3 1 - 4

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Write the augmented matrix for each system and give its dimension. Do not solve. 2x + y + z - 3 = 0 3x - 4y + 2z + 7 = 0 x + y + z - 2 = 0

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In Exercises 13–18, perform each matrix row operation and write the new matrix.

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Write the augmented matrix for each system and give its dimension. Do not solve. 4x - 2y + 3z - 4 = 0 3x + 5y + z - 7 = 0 5x - y + 4z - 7 = 0

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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. A+D

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In Exercises 13–18, perform each matrix row operation and write the new matrix.

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Write the system of equations associated with each augmented matrix . Do not solve. <4x3 Matrix>

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In Exercises 9 - 16, find the following matrices: d. - 3A + 2B 2 - 10 - 2 6 10 - 2 A = 14 12 10 B = 0 - 12 - 4 4 - 2 2 - 5 2 - 2

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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. D-A

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Write the system of equations associated with each augmented matrix . Do not solve. <4x3 Matrix>

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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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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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Write the system of equations associated with each augmented matrix . Do not solve.

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Use the Gauss-Jordan method to solve each system of equations. For systems in two variables with infinitely many solutions, write the solution with y arbitrary. For systems in three variables with infinitely many solutions, write the solution set with z arbitrary. See Examples 1-4. x + y = 5 x - y = -1

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In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

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Solve each system, using the method indicated. 5x + 2y = -10 3x - 5y = -6 (Gauss-Jordan)

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In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.

Solve each system, using the method indicated. 3x + y = -7 x - y = -5 (Gaussian elimination)

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Exercises 57–59 will help you prepare for the material covered in the next section. Solve: A + B = 3, 2A - 2B + C = 17, 4A - 2C =14

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In Exercises 19–22, find the quadratic function y = ax^2+bx+c whose graph passes through the given points. (−1,−4), (1,−2), (2, 5)

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In Exercises 1–4, determine if the given ordered triple is a solution of the system. (4, 1, 2) x−2y=2, 2x+3y=11, y−4z=−7

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In Exercises 1–4, determine if the given ordered triple is a solution of the system. (2,−1, 3) x+ y+0z=4, x−2y−0z=1, 2x−y−2z=−1

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Answer each question. By what expression should we multiply each side of (3x - 1)/(x(2x^2 + 1)^2) = A/x + (Bx + C)/(2x^2 + 1) + (Dx + E)/(2x^2 + 1)^2 so that there are no fractions in the equation?

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Find the partial fraction decomposition for each rational expression. See Examples 1–4. 5/(3x(2x + 1))

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Find the partial fraction decomposition for each rational expression. See Examples 1–4. x/(x^2 + 4x - 5)

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Find the partial fraction decomposition for each rational expression. See Examples 1–4. (4x^2 - x - 15)/(x(x + 1)(x - 1))

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Textbook Question

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Textbook Question

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Textbook Question

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Textbook Question

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Find the partial fraction decomposition for each rational expression. See Examples 1–4. (3x - 1)/(x(2x^2 + 1)^2)

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Find the partial fraction decomposition for each rational expression. See Examples 1–4. (-x^4 - 8x^2 + 3x - 10)/((x + 2)(x^2 + 4)^2)

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Textbook Question

Find the partial fraction decomposition for each rational expression. See Examples 1–4. (x^2)/(x^4 - 1)

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Find the partial fraction decomposition for each rational expression. See Examples 1–4. (4x^2 - 3x - 4)/(x^3 + x^2 - 2x)

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Find the partial fraction decomposition for each rational expression. 5-2x / (x^2 + 2)(x - 1)

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Textbook Question

In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists. w + 2x + 3y - z = 7 2x - 3y + z = 4 w - 4x + y = 3

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Textbook Question

In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists. w + x - y + z = - 2 2w - x + 2y - z = 7 - w + 2x + y + 2z = - 1

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Textbook Question

In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists. x + y - 2z = 2 3x - y - 6z = - 7

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In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists. x + 2y + 3z = 5 y - 5z = 0

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In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists. 2x + y - z = 2 3x + 3y - 2z = 3

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Textbook Question

In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists. w - 3x + y - 4z = 4 - 2w + x + 2y = - 2 3w - 2x + y - 6z = 2 - w + 3x + 2y - z = - 6

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Textbook Question

In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists. 2w + x - y = 3 w - 3x + 2y = - 4 3w + x - 3y + z = 1 w + 2x - 4y - z = - 2

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In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists. w - 2x - y - 3z = - 9 w + x - y = 0 3w + 4x + z = 6 2x - 2y + z = 3

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Textbook Question

In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists. 8x + 5y + 11z = 30 - x - 4y + 2z = 3 2x - y + 5z = 12

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Textbook Question

In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists. 3x + 4y + 2z = 3 4x - 2y - 8z = - 4 x + y - z = 3

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Textbook Question

In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists. 5x + 8y - 6z = 14 3x + 4y - 2z = 8 x + 2y - 2z = 3

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In Exercises 1 - 24, use Gaussian Eliminaion to find the complete solution to each system of equations, or show that none exists. 5x + 12y + z = 10 2x + 5y + 2z = - 1 x + 2y - 3z = 5

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Textbook Question

In Exercises 8–11, use Gaussian elimination to find the complete solution to each system, or show that none exists.

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Textbook Question

In Exercises 8–11, use Gaussian elimination to find the complete solution to each system, or show that none exists.

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Textbook Question

In Exercises 8–11, use Gaussian elimination to find the complete solution to each system, or show that none exists.

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Multiple Choice

Write the equations in standard form, then represent the system using an augmented matrix.

$3x+5y-9=0$

$8x=-4y+3$

Perform the indicated Row Operation.

SWAP $R_1\leftrightarrow R_2$

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Multiple Choice

Perform the indicated Row Operation.

ADD $R_1+2\cdot R_3\rightarrow R_1$

Solve the system of equations by using row operations to write a matrix in REDUCED row-echelon form.

$4x+2y+3z=6$

$x+y+z=3$

$5x+y+2z=5$

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Multiple Choice

Write the system of equations represented by the augmented matrix shown.

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