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Chapter 1: Q1E (page 1)

In Exercises 1, determine which matrices are in reducedechelon form and which others are only in echelon form.


a. \(\left[ {\begin{array}{*{20}{c}}1&0&0&0\\0&1&0&0\\0&0&1&1\end{array}} \right]\)


b. \(\left[ {\begin{array}{*{20}{c}}1&0&1&0\\0&1&1&0\\0&0&0&1\end{array}} \right]\)

c. \(\left[ {\begin{array}{*{20}{c}}1&0&0&0\\0&1&1&0\\0&0&0&0\\0&0&0&1\end{array}} \right]\)

d. \(\left[ {\begin{array}{*{20}{c}}1&1&0&1&1\\0&2&0&2&2\\0&0&0&3&3\\0&0&0&0&4\end{array}} \right]\)

Short Answer

Expert verified

Matrices (a) and (d) are in the reduced echelon form, matrix (c) is not in the echelon form, and matrix (d) is in the echelon form.

Step by step solution

01

Write the conditions for the echelon and reduced echelon forms

Check whether the provided matrix is in the reduced echelon form or just the echelon form for the given augmented matrices.

The matrix is in the echelon form if it satisfies the following conditions:

  • The nonzero rows should be positioned above the zero rows.
  • Each row's leading entry should be in the column to the right of the row above its leading item.
  • In each column, all items below the leading entry should be zero.

For the reduced echelon form, the matrix must follow the above conditions as well as some additional conditions as shown below:

  • Each column's components below the leading entry must be zero.
  • Each column's leading 1 must be the sole nonzero item.
02

Identify if matrix (a) is in the reduced echelon form or echelon form

a.

Consider the matrix \(\left[ {\begin{array}{*{20}{c}}1&0&0&0\\0&1&0&0\\0&0&1&1\end{array}} \right]\).

The following can be observed from the above matrix:

  • The nonzero rows are positioned above the zero rows.
  • Each row's leading entry is in the column to the right of the row above its leading item.
  • In each column, all items below the leading entry are zero.
  • Each column's components below the leading entry are zero.
  • Each column's leading 1 is the sole nonzero item.

Thus, matrix (a) is in the reduced echelon form.

03

Identify if matrix (b) is in the reduced echelon form or echelon form 

b.

Consider the matrix \(\left[ {\begin{array}{*{20}{c}}1&0&1&0\\0&1&1&0\\0&0&0&1\end{array}} \right]\).

The following can be observed from the above matrix:

  • The nonzero rows are positioned above the zero rows.
  • Each row's leading entry is in the column to the right of the row above its leading item.
  • In each column, all items below the leading entry are zero.
  • Each column's components below the leading entry are zero.
  • Each column's leading 1 is the sole nonzero item.

Thus, matrix (b) is in the reduced echelon form.

04

Identify if matrix (c) is in the reduced echelon form or echelon form

c.

Consider the matrix\(\left[ {\begin{array}{*{20}{c}}1&0&0&0\\0&1&1&0\\0&0&0&0\\0&0&0&1\end{array}} \right]\).

The following can be observed from the above matrix:

  • The nonzero rows are not positioned above the zero rows.

Thus, matrix (c) is in the reduced echelon form.

05

Identify if matrix (d) is in the reduced echelon form or echelon form 

d.

Consider the matrix \(\left[ {\begin{array}{*{20}{c}}1&1&0&1&1\\0&2&0&2&2\\0&0&0&3&3\\0&0&0&0&4\end{array}} \right]\).

The following can be observed from the above matrix:

  • The nonzero rows are positioned above the zero rows.
  • Each row's leading entry is in the column to the right of the row above its leading item.
  • In each column, all items below the leading entry are zero.
  • Each column's components below the leading entry are not zero.
  • Each column's leading 1 is not the sole nonzero item.

Thus, matrix (d) is inthe echelon form.

Hence, matrices (a) and (d) are in the reduced echelon form, matrix (c) is not in the echelon form, and matrix (d) is in the echelon form.

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Most popular questions from this chapter

Consider the problem of determining whether the following system of equations is consistent:

\(\begin{aligned}{c}{\bf{4}}{x_1} - {\bf{2}}{x_2} + {\bf{7}}{x_3} = - {\bf{5}}\\{\bf{8}}{x_1} - {\bf{3}}{x_2} + {\bf{10}}{x_3} = - {\bf{3}}\end{aligned}\)

  1. Define appropriate vectors, and restate the problem in terms of linear combinations. Then solve that problem.
  1. Define an appropriate matrix, and restate the problem using the phrase 鈥渃olumns of A.鈥�
  1. Define an appropriate linear transformation T using the matrix in (b), and restate the problem in terms of T.

Consider a dynamical systemwith two components. The accompanying sketch shows the initial state vectorx鈫�0and two eigen vectors蠀1鈫�鈥�鈥�and鈥�鈥�蠀2鈫�of A (with eigen values 位1鈫�and位2鈫�respectively). For the given values of位1鈫�and位2鈫�, draw a rough trajectory. Consider the future and the past of the system.

位1鈫�=0.9,位2鈫�=0.9

Let \({{\mathop{\rm a}\nolimits} _1} = \left[ {\begin{array}{*{20}{c}}1\\4\\{ - 2}\end{array}} \right],{{\mathop{\rm a}\nolimits} _2} = \left[ {\begin{array}{*{20}{c}}{ - 2}\\{ - 3}\\7\end{array}} \right],\) and \({\rm{b = }}\left[ {\begin{array}{*{20}{c}}4\\1\\h\end{array}} \right]\). For what values(s) of \(h\) is \({\mathop{\rm b}\nolimits} \) in the plane spanned by \({{\mathop{\rm a}\nolimits} _1}\) and \({{\mathop{\rm a}\nolimits} _2}\)?

In Exercises 32, find the elementary row operation that transforms the first matrix into the second, and then find the reverse row operation that transforms the second matrix into the first.

32. \(\left[ {\begin{array}{*{20}{c}}1&2&{ - 5}&0\\0&1&{ - 3}&{ - 2}\\0&{ - 3}&9&5\end{array}} \right]\), \(\left[ {\begin{array}{*{20}{c}}1&2&{ - 5}&0\\0&1&{ - 3}&{ - 2}\\0&0&0&{ - 1}\end{array}} \right]\)

Determine whether the statements that follow are true or false, and justify your answer.

15: The systemAx鈬赌=[0001]isinconsistent for all 4脳3 matrices A.
See all solutions

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