Linear Algebra Ch 1.2 — Row Reduction and Echelon Forms

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Linear Algebra Ch 1.2 — Row Reduction and Echelon Forms 0/46
1 Row Reduction and Echelon Forms · Level 1
Determine which matrices are in reduced echelon form and which others are only in echelon form. a. \(\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}\) b. \(\begin{pmatrix} 1 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\) c. \(\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{pmatrix}\) d. \(\begin{pmatrix} 1 & 1 & 0 & 1 & 1 \\ 0 & 2 & 0 & 2 & 2 \\ 0 & 0 & 0 & 3 & 3 \\ 0 & 0 & 0 & 0 & 4 \end{pmatrix}\)
2 Row Reduction and Echelon Forms · Level 1
Determine which matrices are in reduced echelon form and which others are only in echelon form. a. \(\begin{pmatrix} 1 & 1 & 0 & 1 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 \end{pmatrix}\) b. \(\begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}\) c. \(\begin{pmatrix} 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 \end{pmatrix}\) d. \(\begin{pmatrix} 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 2 & 2 & 2 \\ 0 & 0 & 0 & 0 & 3 \\ 0 & 0 & 0 & 0 & 0 \end{pmatrix}\)
3 Row Reduction and Echelon Forms · Level 2
Row reduce the matrix to reduced echelon form. Circle the pivot positions in the final matrix and in the original matrix, and list the pivot columns. \(\begin{pmatrix} 1 & 2 & 3 & 4 \\ 4 & 5 & 6 & 7 \\ 6 & 7 & 8 & 9 \end{pmatrix}\)
4 Row Reduction and Echelon Forms · Level 2
Row reduce the matrix to reduced echelon form. Circle the pivot positions in the final matrix and in the original matrix, and list the pivot columns. \(\begin{pmatrix} 1 & 3 & 5 & 7 \\ 3 & 5 & 7 & 9 \\ 5 & 7 & 9 & 1 \end{pmatrix}\)
5 Row Reduction and Echelon Forms · Level 3
Describe the possible echelon forms of a nonzero \(2 \times 2\) matrix. Use the symbols \(\mathbf{square.filled}\), \(*\), and \(0\), as in the first part of Example 1.
6 Row Reduction and Echelon Forms · Level 3
Repeat Exercise 5 for a nonzero \(3 \times 2\) matrix.
7 Row Reduction and Echelon Forms · Level 2
Find the general solutions of the system whose augmented matrix is given. \(\begin{pmatrix} 1 & 2 & 3 & 4 \\ 4 & 8 & 9 & 4 \end{pmatrix}\)
8 Row Reduction and Echelon Forms · Level 2
Find the general solutions of the system whose augmented matrix is given. \(\begin{pmatrix} 1 & 4 & 0 & 7 \\ 2 & 7 & 0 & 11 \end{pmatrix}\)
9 Row Reduction and Echelon Forms · Level 2
Find the general solutions of the system whose augmented matrix is given. \(\begin{pmatrix} 0 & 1 & -6 & 5 \\ 1 & -2 & 7 & -4 \end{pmatrix}\)
10 Row Reduction and Echelon Forms · Level 2
Find the general solutions of the system whose augmented matrix is given. \(\begin{pmatrix} 1 & -2 & -1 & 3 \\ 3 & -6 & -2 & 2 \end{pmatrix}\)
11 Row Reduction and Echelon Forms · Level 3
Find the general solutions of the system whose augmented matrix is given. \(\begin{pmatrix} 3 & -4 & 2 & 0 \\ -9 & 12 & -6 & 0 \\ -6 & 8 & -4 & 0 \end{pmatrix}\)
12 Row Reduction and Echelon Forms · Level 3
Find the general solutions of the system whose augmented matrix is given. \(\begin{pmatrix} 1 & -7 & 0 & 6 & 5 \\ 0 & 0 & 1 & -2 & -3 \\ -1 & 7 & -4 & 2 & 7 \end{pmatrix}\)
13 Row Reduction and Echelon Forms · Level 3
Find the general solutions of the system whose augmented matrix is given. \(\begin{pmatrix} 1 & -3 & 0 & -1 & 0 & -2 \\ 0 & 1 & 0 & 0 & -4 & 1 \\ 0 & 0 & 0 & 1 & 9 & -4 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\)
14 Row Reduction and Echelon Forms · Level 3
Find the general solutions of the system whose augmented matrix is given. \(\begin{pmatrix} 1 & 2 & -5 & -4 & 0 & -5 \\ 0 & 1 & -6 & -4 & 0 & 2 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{pmatrix}\)
15 Row Reduction and Echelon Forms · Level 2
Write down the equations corresponding to the augmented matrix in Exercise 9 and verify that your answer to Exercise 9 is correct by substituting the solutions you obtained back into the original equations.
16 Row Reduction and Echelon Forms · Level 2
Write down the equations corresponding to the augmented matrix in Exercise 10 and verify that your answer to Exercise 10 is correct by substituting the solutions you obtained back into the original equations.
17 Row Reduction and Echelon Forms · Level 2
Write down the equations corresponding to the augmented matrix in Exercise 11 and verify that your answer to Exercise 11 is correct by substituting the solutions you obtained back into the original equations.
18 Row Reduction and Echelon Forms · Level 2
Write down the equations corresponding to the augmented matrix in Exercise 12 and verify that your answer to Exercise 12 is correct by substituting the solutions you obtained back into the original equations.
19 Row Reduction and Echelon Forms · Level 3
Use the notation of Example 1 for matrices in echelon form. Suppose each matrix represents the augmented matrix for a system of linear equations. In each case, determine if the system is consistent. If the system is consistent, determine if the solution is unique. a. \(\begin{pmatrix} \mathbf{square.filled} & * & * & * \\ 0 & \mathbf{square.filled} & * & * \\ 0 & 0 & \mathbf{square.filled} & 0 \end{pmatrix}\) b. \(\begin{pmatrix} 0 & \mathbf{square.filled} & * & * & * \\ 0 & 0 & \mathbf{square.filled} & * & * \\ 0 & 0 & 0 & 0 & \mathbf{square.filled} \end{pmatrix}\)
20 Row Reduction and Echelon Forms · Level 3
Use the notation of Example 1 for matrices in echelon form. Suppose each matrix represents the augmented matrix for a system of linear equations. In each case, determine if the system is consistent. If the system is consistent, determine if the solution is unique. a. \(\begin{pmatrix} \mathbf{square.filled} & * & * \\ 0 & \mathbf{square.filled} & * \\ 0 & 0 & 0 \end{pmatrix}\) b. \(\begin{pmatrix} \mathbf{square.filled} & * & * & * & * \\ 0 & 0 & \mathbf{square.filled} & * & * \\ 0 & 0 & 0 & \mathbf{square.filled} & * \end{pmatrix}\)
21 Row Reduction and Echelon Forms · Level 3
Determine the value(s) of \(h\) such that the matrix is the augmented matrix of a consistent linear system. \(\begin{pmatrix} 2 & 3 & h \\ 4 & 6 & 7 \end{pmatrix}\)
22 Row Reduction and Echelon Forms · Level 3
Determine the value(s) of \(h\) such that the matrix is the augmented matrix of a consistent linear system. \(\begin{pmatrix} 1 & -4 & -3 \\ 6 & h & -9 \end{pmatrix}\)
23 Row Reduction and Echelon Forms · Level 3
Choose \(h\) and \(k\) such that the system has (a) no solution, (b) a unique solution, and (c) many solutions. Give separate answers for each part. \(x_1 + h x_2 = 2\) \(4 x_1 + 8 x_2 = k\)
24 Row Reduction and Echelon Forms · Level 3
Choose \(h\) and \(k\) such that the system has (a) no solution, (b) a unique solution, and (c) many solutions. Give separate answers for each part. \(x_1 + 4 x_2 = 5\) \(2 x_1 + h x_2 = k\)
25 Row Reduction and Echelon Forms · Level 2
(T/F) In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.
26 Row Reduction and Echelon Forms · Level 2
(T/F) The echelon form of a matrix is unique.
27 Row Reduction and Echelon Forms · Level 2
(T/F) The row reduction algorithm applies only to augmented matrices for a linear system.
28 Row Reduction and Echelon Forms · Level 2
(T/F) The pivot positions in a matrix depend on whether row interchanges are used in the reduction process.
29 Row Reduction and Echelon Forms · Level 2
(T/F) A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.
30 Row Reduction and Echelon Forms · Level 2
(T/F) Reducing a matrix to echelon form is called the forward phase of the row reduction process.
31 Row Reduction and Echelon Forms · Level 2
(T/F) Finding a parametric description of the solution set of a linear system is the same as solving the system.
32 Row Reduction and Echelon Forms · Level 2
(T/F) Whenever a system has free variables, the solution set contains a unique solution.
33 Row Reduction and Echelon Forms · Level 2
(T/F) If one row in an echelon form of an augmented matrix is \([0 \ 0 \ 0 \ 0 \ 5]\), then the associated linear system is inconsistent.
34 Row Reduction and Echelon Forms · Level 2
(T/F) A general solution of a system is an explicit description of all solutions of the system.
35 Row Reduction and Echelon Forms · Level 3
Suppose a \(3 \times 5\) coefficient matrix for a system has three pivot columns. Is the system consistent? Why or why not?
36 Row Reduction and Echelon Forms · Level 3
Suppose a system of linear equations has a \(3 \times 5\) augmented matrix whose fifth column is a pivot column. Is the system consistent? Why (or why not)?
37 Row Reduction and Echelon Forms · Level 3
Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.
38 Row Reduction and Echelon Forms · Level 3
Suppose the coefficient matrix of a linear system of three equations in three variables has a pivot in each column. Explain why the system has a unique solution.
39 Row Reduction and Echelon Forms · Level 3
Restate the last sentence in Theorem 2 using the concept of pivot columns: "If a linear system is consistent, then the solution is unique if and only if _______."
40 Row Reduction and Echelon Forms · Level 3
What would you have to know about the pivot columns in an augmented matrix in order to know that the linear system is consistent and has a unique solution?
41 Row Reduction and Echelon Forms · Level 3
A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system. Suppose that such a system happens to be consistent. Explain why there must be an infinite number of solutions.
42 Row Reduction and Echelon Forms · Level 3
Give an example of an inconsistent underdetermined system of two equations in three unknowns.
43 Row Reduction and Echelon Forms · Level 3
A system of linear equations with more equations than unknowns is sometimes called an overdetermined system. Can such a system be consistent? Illustrate your answer with a specific system of three equations in two unknowns.
44 Row Reduction and Echelon Forms · Level 3
Suppose an \(n \times (n + 1)\) matrix is row reduced to reduced echelon form. Approximately what fraction of the total number of operations (flops) is involved in the backward phase of the reduction when \(n = 30\)? when \(n = 300\)?
45 Row Reduction and Echelon Forms · Level 4
Find the interpolating polynomial \(p(t) = a_0 + a_1 t + a_2 t^2\) for the data \((1, 11)\), \((2, 16)\), \((3, 19)\). That is, find \(a_0\), \(a_1\), and \(a_2\) such that \(a_0 + a_1 (1) + a_2 (1)^2 = 11\) \(a_0 + a_1 (2) + a_2 (2)^2 = 16\) \(a_0 + a_1 (3) + a_2 (3)^2 = 19\)
46 Row Reduction and Echelon Forms · Level 5
In a wind tunnel experiment, the force on a projectile due to air resistance was measured at different velocities: Velocity (100 ft/sec): 0, 2, 4, 6, 8, 10 Force (100 lb): 0, 2.90, 14.8, 39.6, 74.3, 119 Find an interpolating polynomial for these data and estimate the force on the projectile when the projectile is traveling at 750 ft/sec. Use \(p(t) = a_0 + a_1 t + a_2 t^2 + a_3 t^3 + a_4 t^4 + a_5 t^5\). What happens if you try to use a polynomial of degree less than 5? (Try a cubic polynomial, for instance.)

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